gravitation6-214bbc04-e7e9-47a7-95a9-eb77e413b5ce

снартев 22

THERMODYNAMICS, HYDRODYNAMICS, ELECTRODYNAMICS, GEOMETRIC OPTICS, AND KINETIC THEORY

§22.1. THE WHY OF THIS CHAPTER

Astrophysical applications of gravitation theory are the focus of the rest of this book, except for Chapters 41-44. Each application-stars, star clusters, cosmology, collapse, black holes, gravitational waves, solar-system experiments-can be pursued by itself at an elementary level, without reference to the material in this chapter. But deep understanding of the applications requires a prior grasp of thermodynamics, hydrodynamics, electrodynamics, geometric optics, and kinetic theory, all in the context of curved spacetime. Hence, most Track-2 readers will want to probe these subjects at this point.

§22.2. THERMODYNAMICS IN CURVED SPACETIME*

Consider, for concreteness and simplicity, the equilibrium thermodynamics of a perfect fluid with fixed chemical composition ("simple perfect fluid")-for example, the gaseous interior of a collapsing supermassive star. The thermodynamic state of a fluid element, as it passes through an event

P_{0}

, can be characterized by various thermodynamic potentials, such as

n, ρ, p, T, s, μ

. The numerical value of each potential at

P_{0}

is measured in the proper reference frame (

$ 13.6

) of an observer who moves with the fluid element-i.e., in the fluid element's "rest frame." Despite

This chapter is entirely Track 2. No earlier Track-2 material is needed as preparation for it, but Chapter 5 (stress-energy tensor) will be helpful.
§22.5 (geometric optics) is needed as preparation for Chapter 34 (singularities and global methods). The rest of the chapter is not needed as preparation for any later chapter; but it will be extremely helpful in most applications of gravitation theory (Chapters 23-40).

Thermodynamic potentials are defined in rest frame of fluid

Definitions of thermodynamic potentials

Definition of "simple fluid"

Law of baryon conservation
this use of rest frame to measure the potentials, the potentials are frame-independent functions (scalar fields). At the chosen event

P_{0}

, a given potential (e.g.,

n

) has a unique value

n (P_{0})

; so

n

is a perfectly good frame-independent function.

The values of

n, ρ, p, T, s, μ

measure the following quantities in the rest frame of the fluid element:

n

, baryon number density; i.e., number of baryons per unit three-dimensional volume of rest frame, with antibaryons (if any) counted negatively.

ρ

, density of total mass-energy; i.e., total mass-energy (including rest mass, thermal energy, compressional energy, etc.) contained in a unit three-dimensional volume of the rest frame.

p

, isotropic pressure in rest frame.

T

, temperature in rest frame.

s

, entropy per baryon in rest frame. (The entropy per unit volume is

n s

μ

, chemical potential of baryons in rest frame [see equation (22.8) below].

The chemical composition of the fluid (number density of hydrogen molecules, number density of hydrogen atoms, number density of free protons and electrons, number density of photons, number density of

^{238} U

nuclei, number density of

Λ

hyperons . . .) is assumed to be fixed uniquely by two thermodynamic variables-e.g., by the total number density of baryons

n

and the entropy per baryon

s

. In this sense the fluid is a "simple fluid." Simple fluids occur whenever the chemical abundances are "frozen" (reaction rates too slow to be important on the time scales of interest; for example, in a supermassive star except during explosive burning and except at temperatures high enough for

e^{-} - e^{+}

pair production). Simple fluids also occur in the opposite extreme of complete chemical equilibrium (reaction rates fast enough to maintain equilibrium despite changing density and entropy; for example, in neutron stars, where high pressures speed up all reactions). When one examines nuclear burning in a nonconvecting star, or explosive nuclear burning, or pair production and neutrino energy losses at high temperatures, one must usually treat the fluid as "multicomponent." Then one introduces a number density

n_{J}

and a chemical potential

μ_{J}

for each chemical species with abundance not fixed by

n

and

s

. For further details see, e.g., Zel'dovich and Novikov (1971).

The most fundamental law of thermodynamics-even more fundamental than the "first" and "second" laws-is baryon conservation. Consider a fluid element whose moving walls are attached to the fluid so that no baryons flow in or out. As the fluid element moves through spacetime, deforming along the way, its volume

V

changes. But the number of baryons in it must remain fixed, so

\begin{matrix} (22.1) & \frac{d}{d τ} (n V) = 0 \end{matrix}

The changes in volume are produced by the flow of neighboring bits of fluid away from or toward each other-explicitly (exercise 22.1)

\begin{matrix} (22.2) & d V / d τ = (\nabla \cdot u) V \end{matrix}

where

u = d / d τ

is the 4 -velocity of the fluid. Consequently, baryon conservation [equation (22.1)] can be reexpressed as

\begin{aligned} 0 = \frac{d n}{d τ} + \frac{n}{V} \frac{d V}{d τ} = \nabla_{u} n + n (\nabla \cdot u) = u \cdot \nabla n + n (\nabla \cdot u) = \nabla \cdot (n u); \\ i.e., \\ (22.3) & \nabla \cdot S = 0, \\ (22.4) & S = n u = baryon number-flux vector \end{aligned}

(see

§ 5.4

and exercise 5.3.) Moreover, this abstract geometric version of the law must be just as valid in curved spacetime as in flat (equivalence principle).

Note the analogy with the law of charge conservation,

\nabla \cdot J = 0

, in electrodynamics (exercise 3.16) and with the local law of energy-momentum conservation,

\nabla \cdot T = 0 ($ 85.9

and 16.2). In a very deep sense, the forms of these three laws are dictated by the theorem of Gauss (

$ 5.9

, and Boxes 5.3, 5.4).

The second law of thermodynamics states that, in flat spacetime or in curved, entropy can be generated but not destroyed. Apply this law to a fluid element of volume

V

containing a fixed number of baryons

N

. The entropy it contains is

S = N s = n s V

Entropy may flow in and out across the faces of the fluid element ("heat flow" between neighboring fluid elements); but for simplicity assume it does not; or if it does, assume that it flows too slowly to have any significance for the problem at hand. Then the entropy in the fluid element can only increase:

\begin{array}{ll} d (n s V) / d τ \geq 0 & when negligible entropy is exchanged between \\ neighboring fluid elements; \end{array}

i.e. [combine with equation (22.1)]

\begin{matrix} (22.5) & d s / d τ \geq 0 (no entropy exchange). \end{matrix}

So long as the fluid element remains in thermodynamic equilibrium, its entropy will actually be conserved ["

=

" in equation (22.5)]; but at a shock wave, where equilibrium is momentarily broken, the entropy will increase (conversion of "relative kinetic energy" of neighboring fluid elements into heat). [For discussions of heat flow in special and general relativity, see Exercise 22.7. For discussion of shock waves, see Taub (1948), de Hoffman and Teller (1950), Israel (1960), May and White (1967), Zel'dovich and Rayzer (1967), Lichnerowicz (1967, 1971), and Thorne (1973a).]

The first law of thermodynamics, in the proper reference frame of a fluid element, Shock waves and heat flow First law of thermodynamics is identical to the first law in flat spacetime ("principle of equivalence"); and in flat spacetime the first law is merely the law of energy conservation:

d (\binom{energy in a volume element containing}{a fixed number, A, of baryons}) = - p d (volume) + T d (entropy);

Second law of thermodynamics

i.e.,

d (ρ A / n) = - p d (A / n) + T d (A s)

i.e.,

d ρ = \frac{ρ + p}{n} d n + n T d s

Query: what kind of a "d" appears here? For a simple fluid, the values of two potentials, e.g.,

n

and

s

, fix all the others uniquely; so any change in

ρ

must be determined uniquely by the changes in

n

and

s

. It matters not whether the changes are measured along the world line of a given fluid element, or in some other direction. Thus, the "

d

" in the first law can be interpreted as an exterior derivative

\begin{matrix} (22.6) & d ρ = \frac{ρ + p}{n} d n + n T d s; \end{matrix}

and the changes along a given direction in the fluid (along a given tangent vector

v

) can be written

\begin{aligned} \nabla_{v} ρ & \equiv ⟨ d ρ, v ⟩ = \frac{ρ + p}{n} ⟨ d n, v ⟩ + n T ⟨ d s, v ⟩ \\ = \frac{ρ + p}{n} \nabla_{v} n + n T \nabla_{v} s . \end{aligned}

Equation (22.6) lends itself to interpretation in two opposite senses: as a way to deduce the density of mass-energy of the medium from information about pressure (as a function of

n

and

s

) and temperature (as a function of

n

and

s

); and conversely, as a way to deduce the two functions

p (n, s)

and

T (n, s)

from the one function

ρ (n, s)

. It is natural to look at the second approach first; who does not like a strategy that makes an intellectual profit? Regarding

ρ

as a known (or calculable) function of

n

and

s

, one deduces from (22.6)

\begin{matrix} \frac{ρ + p}{n} = {(\frac{\partial ρ}{\partial n})}_{s} \\ n T = {(\frac{\partial ρ}{\partial s})}_{n} \end{matrix}

and thence pressure and temperature individually,

\begin{matrix} (22.7a) & p (n, s) = n {(\frac{\partial ρ}{\partial n})}_{s} - ρ \\ (22.7b) & T (n, s) = \frac{1}{n} {(\frac{\partial ρ}{\partial s})}_{n} \end{matrix}

("two equations of state from one"). The analysis simplifies still further when the fluid, already assumed to be everywhere of the same composition, is also everywhere
endowed with the same entropy per baryon,

s

, and is in a state of adiabatic flow (no shocks or heat conduction). Then the density

ρ = ρ (n, s)

reduces to a function of one variable out of which one derives everything (

ρ, p, μ

) needed for the hydrodynamics and the gravitation physics of the system (next chapter). Other choices of the "primary thermodynamic potential" are appropriate under other circumstances (see Box 22.1).

If differentiation leads from

ρ (n, s)

p (n, s)

and

T (n, s)

, it does not follow that one can take any two functions

p (n, s)

and

T (n, s)

and proceed "backwards" (by integration) to the "primary function",

ρ (n, s)

. To be compatible with the first law of thermodynamics (22.6), the two functions must satisfy the consistency requirement ["Maxwell relation"; equality of second partial derivatives of

ρ

]

Maxwell relation

\begin{matrix} (22.7c) & (\partial p / \partial s)_{n} = n^{2} (\partial T / \partial n)_{s} . \end{matrix}

Box 22.1 PRINCIPAL ALTERNATIVES FOR "PRIMARY THERMODYNAMIC POTENTIAL" TO DESCRIBE A FLUID

Primary thermodynamic potential and quantities on which it is most appropriately envisaged to depend
"Secondary" thermodynamic quantities obtained by differentiation of primary with or without use of

d (\frac{ρ}{n}) + p d (\frac{1}{n}) - T d s = 0

"Density"; total amount of massenergy (rest + thermal

+ \dots

) per unit volume

p (n, s) = n {(\frac{\partial ρ}{\partial n})}_{s} - ρ

ρ = ρ (n, s) T (n, s) = \frac{1}{n} {(\frac{\partial ρ}{\partial s})}_{n},

"Physical free energy"

a (n, T) = \frac{ρ}{n} - T s

\begin{aligned} p (n, T) & = n^{2} {(\frac{\partial a}{\partial n})}_{T} \\ s (n, T) & = - {(\frac{\partial a}{\partial T})}_{n} \\ ρ (n, T) & = - n T^{2} {[\frac{\partial (a / T)}{\partial T}]}_{n} \end{aligned}

"Chemical free energy"

f (p, T) = \frac{ρ + p}{n} - T s

\begin{aligned} 1 / n (p, T) & = (\partial f / \partial p)_{T} \\ s (p, T) & = - (\partial f / \partial T)_{p} \\ ρ (p, T) & = \frac{f - T (\partial f / \partial T)_{p}}{(\partial f / \partial p)_{T}} - p \end{aligned}

"Chemical potential" ("energy to inject" expressed on a "per baryon" basis)

μ (p, s) = \frac{p + ρ}{n}

\begin{aligned} 1 / n (p, s) & = (\partial μ / \partial p)_{s} \\ T (p, s) & = (\partial μ / \partial s)_{p} \\ ρ (p, s) & = \frac{μ}{(\partial μ / \partial p)_{s}} - p \end{aligned}

Conditions under which convenient, appropriate, and relevant

Chemical potential equals
"injection energy" at fixed entropy per baryon and total volume

Laws of hydrodynamics for simple fluid without heat flow or viscosity:

The chemical potential

μ

is also a unique function of

n

and

s

. It is defined as follows. (1) Take a sample of the simple fluid in a fixed thermodynamic state (fixed

n

and

s

). (2) Take, separately, a much smaller sample of the same fluid, containing

δ A

baryons in the same thermodynamic state as the large sample (same

n

and

s

). (3) Inject the smaller sample into the larger one, holding the volume of the large sample fixed during the injection process. (4) The total mass-energy injected,

δ M_{injected} = ρ \times (volume of injected fluid) = ρ (δ A / n),

plus the work required to perform the injection

\begin{aligned} δ W_{injection} & = (\binom{work done against pressure of large sample}{to open up space in it for the injected fluid}) \\ = p (volume of injected fluid) = p (δ A / n), \end{aligned}

is equal to

μ δ A

μ δ A = δ M_{injected} + δ W_{injection} = \frac{ρ + p}{n} δ A .

Stated more briefly:

\begin{aligned} (22.8) & μ & = (\begin{matrix} total mass-energy required, per baryon, to "create" and \\ inject a small additional amount of fluid into a given \\ sample, without changing s or volume of the sample \end{matrix}) \\ = \frac{ρ + p}{n} = {(\frac{\partial ρ}{\partial n})}_{s} . \\ [by first law of thermodynamics (22.6)] \end{aligned}

All the above laws and equations of thermodynamics are the same in curved spacetime as in flat spacetime; and the same in (relativistic) flat spacetime as in classical nonrelativistic thermodynamics-except for the inclusion of rest mass, together with all other forms of mass-energy, in

ρ

and

μ

. The reason is simple: the laws are all formulated as scalar equations linking thermodynamic variables that one measures in the rest frame of the fluid.

§22.3. HYDRODYNAMICS IN CURVED SPACETIME*

A simple perfect fluid flows through spacetime. It might be the Earth's atmosphere circulating in the Earth's gravitational field. It might be the gaseous interior of the Sun at rest in its own gravitational field. It might be interstellar gas accreting onto a black hole. But whatever and wherever the fluid may be, its motion will be governed by the curved-spacetime laws of thermodynamics (

$ 22.2

) plus the local

law of energy-momentum conservation,

\nabla \cdot T = 0

. The chief objective of this section is to reduce the equation

\nabla \cdot T = 0

to usable form. The reduction will be performed in the text using abstract notation; the reader is encouraged to repeat the reduction using index notation.
The stress-energy tensor for a perfect fluid, in curved spacetime as in flat (equivalence principle!), is

\begin{matrix} (22.9) & T = (ρ + p) u \otimes u + p g \end{matrix}

(See §5.5.) Its divergence is readily calculated using the chain rule; using the compatibility relation between

g

and

\nabla, \nabla g = 0

; using the identity

(\nabla p) \cdot g = \nabla p

(which one readily verifies in index notation); and using

\begin{aligned} 0 & = \nabla \cdot T = [\nabla (ρ + p) \cdot u] u + [(ρ + p) \nabla \cdot u] u + [(ρ + p) u] \cdot \nabla u + (\nabla p) \cdot g \\ Q_{[divergence on first slot]} \\ (22.10) & = [\nabla_{u} ρ + \nabla_{u} p + (ρ + p) \nabla \cdot u] u + (ρ + p) \nabla_{u} u + \nabla^{'} p . \end{aligned}

The component of this equation along the 4 -velocity is especially simple (recall that

u \cdot \nabla_{u} u = \frac{1}{2} \nabla_{u} u^{2} = 0

because

u^{2} \equiv - 1

\begin{aligned} 0 & = u \cdot (\nabla \cdot T) = - [\nabla_{u} ρ + \nabla_{u} p + (ρ + p) \nabla \cdot u] + \nabla_{u} p \\ = - \nabla_{u} ρ - (ρ + p) \nabla \cdot u . \end{aligned}

Combine this with the equation of baryon conservation (22.3); the result is

\begin{matrix} (22.11a) & \frac{d ρ}{d τ} = \frac{(ρ + p)}{n} \frac{d n}{d τ} \end{matrix}

(2) Local energy conservation: adiabaticity of flow

Notice that this is identical to the first law of thermodynamics (22.6) applied along a flow line, plus the assumption that the entropy per baryon is conserved along a flow line

\begin{matrix} (22.11b) & d s / d τ = 0 \end{matrix}

There is no reason for surprise at this result. To insist on thermodynamic equilibrium and to demand that the entropy remain constant is to require zero exchange of heat between one element of the fluid and another. But the stress-energy tensor (22.9) recognizes that heat exchange is absent. Any heat exchange would show up as an energy flux term in

T

(Ex. 22.7); but no such term is present. Consequently, when one studies local energy conservation by evaluating

u \cdot (\nabla \cdot T) = 0

, the stress-energy tensor reports that no heat flow is occurring-i.e. that

d s / d τ = 0

Three components of

\nabla \cdot T = 0

remain: the components orthogonal to the fluid's 4 -velocity. One can pluck them out of

\nabla \cdot T = 0

, leaving behind the component along

u

, by use of the "projection tensor"

\begin{matrix} (22.12) & P \equiv g + u \otimes u \end{matrix}

Box 22.2 THERMODYNAMICS AND HYDRODYNAMICS FOR A SIMPLE PERFECT FLUID IN CURVED SPACETIME

A. Ten Quantities Characterize the Fluid

Thermodynamic potentials all measured in rest frame

n

, baryon number density

ρ

, density of total mass-energy

p

, pressure

T

, temperature

s

, entropy per baryon

μ

, chemical potential per baryon
Four components of the fluid 4-velocity
B. Ten Equations Govern the Fluid's Motion

Two equations of state

\begin{matrix} (1) & p = p (n, s), T = T (n, s) \end{matrix}

subject to the compatibility constraint ("Maxwell relation," which follows from first law of thermodynamics)

(\partial p / \partial s)_{n} = n^{2} (\partial T / \partial n)_{s}

First law of thermodynamics

\begin{matrix} (3) & d ρ = \frac{ρ + p}{n} d n + n T d s \end{matrix}

which can be integrated to give

ρ (n, s)

Equation for chemical potential

\begin{matrix} (4) & μ = (ρ + p) / n \end{matrix}

which can be combined with

ρ (n, s)

and

p (n, s)

to give

μ (n, s)

.
Law of baryon conservation

\begin{matrix} (5) & d n / d τ \equiv \nabla_{u} n = - n \nabla \cdot u \end{matrix}

Conservation of energy along flow lines, which (assuming no energy exchange between adjacent fluid elements) means "adiabatic flow"

d s / d τ = 0

except in shock waves, where

d s / d τ > 0

.
[Shock waves are not treated in this book; see Taub (1948), de Hoffman and Teller (1950), Israel (1960), May and White (1967), Zel'dovich and Rayzer (1967); Lichnerowicz (1967, 1971); and Thorne (1973a).]
Euler equations

\begin{matrix} (7) & (ρ + p) \nabla_{u} u = - (g + u \otimes u) \cdot \nabla p \end{matrix}

which determine the flow lines to which

u

is tangent.
Normalization of 4-velocity

\begin{matrix} (10) & u \cdot u = - 1 \end{matrix}

(3) Euler equation
(See exercise 22.4.) Contracting

P

with

\nabla \cdot T = 0

[equation (22.10)] gives

\begin{matrix} (22.13) & (ρ + p) \nabla_{u} u = - P \cdot (\nabla p) \equiv - [\nabla p + (\nabla_{u} p) u] \end{matrix}

This is the "Euler equation" of relativistic hydrodynamics. It has precisely the same form as the corresponding flat-spacetime Euler equation:

(\begin{array}{l} inertial mass \\ per unit volume \\ [exercise 5.4] \end{array}) \times (\binom{4 -acceleration}{of fluid}) = - (\begin{array}{l} pressure gradient \\ in the 3-surface \\ orthogonal to 4-velocity \end{array})

.
The pressure gradient, not "gravity," is responsible for all deviation of flow lines from geodesics.

Box 22.2 reorganizes and summarizes the above laws of thermodynamics and hydrodynamics.

Exercise 22.1. DIVERGENCE OF FLOW LINES PRODUCES VOLUME CHANGES

EXERCISES

Derive the equation

d V / d τ = (\nabla \cdot u) V

[equation (22.2)] for the rate of change of volume of a fluid element. [Hint: Pick an event

P_{0}

, and calculate in a local Lorentz frame at

P_{0}

which momentarily moves with the fluid ("rest frame at

P_{0}

").] [Solution: At events near

P_{0}

the fluid has a very small ordinary velocity

v^{j} = d x^{j} / d t

. Consequently a cube of fluid at

P_{0}

with edges

Δ x = Δ y = Δ z = L

changes its edges, after time

δ t

, by the amounts

\begin{aligned} δ (Δ x) & = [(d x / d t) δ t]_{at} "front face" - [(d x / d t) δ t]_{at} "back face" \\ = (\partial v^{x} / \partial x) L δ t, \\ δ (Δ y) & = (\partial v^{y} / \partial y) L δ t, \\ δ (Δ z) & = (\partial v^{z} / \partial z) L δ t . \end{aligned}

The corresponding change in volume is

δ (Δ x Δ y Δ z) = (\partial v^{j} / \partial x^{j}) L^{3} δ t

so the rate of change of volume is

\partial V / \partial t = V (\partial v^{j} / \partial x^{j})

But in the local Lorentz rest frame at and near

P_{0}

(where

x^{α} = 0

), the metric coefficients are

g_{μ ν} = η_{μ ν} + 0 ({| x^{α} |}^{2})

, and the ordinary velocity is

v^{j} = 0 (| x^{α} |)

; so

\begin{matrix} u^{0} = \frac{d t}{d τ} = \frac{d t}{{(- g_{μ ν} d x^{μ} d x^{ν})}^{1 / 2}} = 1 + 0 ({| x^{α} |}^{2}) \\ u^{j} = \frac{d x^{j}}{d τ} = v^{j} + 0 (∣ x^{α ∣ 3}) \end{matrix}

Thus, the derivatives

\partial V / \partial t

and

V (\partial v^{j} / \partial x^{j})

P_{0}

are

\begin{aligned} \partial V / \partial t & = u^{α} \partial V / \partial x^{α} = u^{α} V_{, α} = d V / d τ \\ = V (\partial v^{j} / \partial x^{j}) = V (\partial u^{α} / \partial x^{α}) = V u^{α}_{; α} = V (\nabla \cdot u) . Q.E.D.] \end{aligned}

[Note that by working in flat spacetime, one could have inferred more easily that

\partial V / \partial t =

d V / d τ

and

\partial v^{j} / \partial x^{j} = \nabla \cdot u

; one would then have concluded

d V / d τ = (\nabla \cdot u) V

; and one could have invoked the equivalence principle to move this law into curved spacetime.]

Exercise 22.2. EQUATION OF CONTINUITY

Show that in the nonrelativistic limit in flat spacetime the equation of baryon conservation (22.3) becomes the "equation of continuity"

\frac{\partial n}{\partial t} + \frac{\partial}{\partial x^{j}} (n v^{j}) = 0

Exercise 22.3. CHEMICAL POTENTIAL FOR IDEAL FERMI GAS

Show that the chemical potential of an ideal Fermi gas, nonrelativistic or relativistic, is (at zero temperature) equal to the Fermi energy (energy of highest occupied momentum state) of that gas.

Exercise 22.4. PROJECTION TENSORS

Show that contraction of a tangent vector

B

with the "projection tensor"

P \equiv g + u \otimes u

projects

B

into the 3 -surface orthogonal to the 4 -velocity vector

u

. [Hint: perform the
calculation in an orthonormal frame with

e_{\hat{0}} = u

, and write

B = B^{α} e_{\hat{α}}

; then show that

P \cdot B = B^{j} e_{j . I}

. If

n

is a unit spacelike vector, show that

P \equiv g - n \otimes n

is the corresponding projection operator. Note: There is no unique concept of "the projection orthogonal to a null vector." Why? [Hint: draw pictures in flat spacetime suppressing one spatial dimension.]

Exercise 22.5. PRESSURE GRADIENT IN STATIONARY GRAVITATIONAL FIELD

A perfect fluid is at rest (flow lines have

x^{j} =

constant) in a stationary gravitational field (metric coefficients are independent of

x^{0}

). Show that the pressure gradient required to "support the fluid against gravity" (i.e., to make its flow lines be

x^{j} =

constant instead of geodesics) is

\begin{matrix} (22.14) & \frac{\partial p}{\partial x^{0}} = 0, \frac{\partial p}{\partial x^{j}} = - (ρ + p) \frac{\partial \ln \sqrt{- g_{00}}}{\partial x^{j}} \end{matrix}

Evaluate this pressure gradient in the Newtonian limit, using the coordinate system and metric coefficients of equation ( 18.15 c ).

Exercise 22.6. EXPANSION, ROTATION, AND SHEAR

Let a field of fluid 4 -velocities

u (P)

be given.
(a) Show that

\nabla u

can be decomposed in the following manner:

\begin{matrix} (22.15a) & u_{α; β} = ω_{α β} + σ_{α β} + \frac{1}{3} θ P_{α β} - a_{α} u_{β} \end{matrix}

where

a

is the 4-acceleration of the fluid

\begin{matrix} (22.15b) & a_{α} \equiv u_{α; β} u^{β} \end{matrix}

θ

is the "expansion" of the fluid world lines

\begin{matrix} (22.15c) & θ \equiv \nabla \cdot u = u_{; α}^{α} \end{matrix}

P_{α β}

is the projection tensor

\begin{matrix} (22.15d) & P_{α β} \equiv g_{α β} + u_{α} u_{β} \end{matrix}

σ_{α β}

is the shear tensor of the fluid

\begin{matrix} (22.15e) & σ_{α β} \equiv \frac{1}{2} (u_{α; μ} P_{β}^{μ} + u_{β; μ} P_{α}^{μ}) - \frac{1}{3} θ P_{α β} \end{matrix}

and

ω_{α β}

is the rotation 2-form of the fluid

\begin{matrix} (22.15f) & ω_{α β} \equiv \frac{1}{2} (u_{α; μ} P_{β}^{μ} - u_{β; μ} P_{α}^{μ}) \end{matrix}

(b) Each of the component parts of this decomposition has a simple physical interpretation in the local rest frames of the fluid. The interpretation of the 4-acceleration

a

in terms of accelerometer readings should be familiar. Exercise 22.1 showed that the expansion

θ = \nabla \cdot u

describes the rate of increase of the volume of a fluid element,

\begin{matrix} (22.15~g) & θ = (1 / V) (d V / d τ) \end{matrix}

Exercise 22.4 explored the meaning and use of the projection tensor

P

. Verify that in a local Lorentz frame (

g_{\hat{α} \hat{β}} = η_{α β}, Γ^{\hat{α}}_{\hat{β} \hat{γ}} = 0

) momentarily moving with the fluid (

u^{\hat{α}} = δ^{α}_{0}

σ_{\hat{α} \hat{β}}

and

ω_{\dot{α} \hat{β}}

reduce to the classical (nonrelativistic) shear and rotation of the fluid. [See, e.g.,

$ 82.4

and 2.5 of Ellis (1971) for both classical and relativistic descriptions of shear and rotation.]

Exercise 22.7. HYDRODYNAMICS WITH VISCOSITY AND HEAT FLOW.*

(a) In

§ 15

of Landau and Lifshitz (1959), one finds an analysis of viscous stresses for a classical (nonrelativistic) fluid. By carrying that analysis over directly to the local Lorentz rest frame of a relativistic fluid, and by then generalizing to frame-independent language, show that the contribution of viscosity to the stress-energy tensor is

\begin{matrix} (22.16a) & T^{(visc)} = - 2 η σ - ζ θ P \end{matrix}

where

η \geq 0

is the "coefficient of dynamic viscosity";

ζ \geq 0

is the "coefficient of bulk viscosity"; and

σ, θ, P

are the shear, expansion, and projection tensor of the fluid.
(b) An idealized description of heat flow in a fluid introduces the heat-flux 4-vector

q

with components in the local rest-frame of the fluid,

\begin{matrix} (22.16b) & q^{\hat{0}} = 0, q^{\hat{j}} = (\binom{energy per unit time crossing unit}{surface perpendicular to e_{3}}) \end{matrix}

By generalizing from the fluid rest frame to frame-independent language, show that the contribution of heat flux to the stress-energy tensor is

\begin{matrix} (22.16c) & T^{(heat)} = u \otimes q + q \otimes u \end{matrix}

Thereby conclude that, in this idealized picture, the stress-energy tensor for a viscous fluid with heat conduction is

\begin{matrix} (22.16d) & T^{α β} = ρ u^{α} u^{β} + (p - ζ θ) P^{α β} - 2 η σ^{α β} + q^{α} u^{β} + u^{α} q^{β} \end{matrix}

s

\begin{matrix} (22.16e) & s \equiv n s u + q / T \end{matrix}

By calculations in the local rest-frame of the fluid, show that

\begin{aligned} \nabla \cdot s & = (\binom{rate of increase of entropy}{in a unit volume}) - (\binom{rate at which heat and fluid}{carry entropy into a unit volume}) \\ (22.16f) & = (\binom{rate at which entropy is being}{generated in a unit volume}) \end{aligned}

Thereby arrive at the following form of the second law of thermodynamics:

\begin{matrix} (22.16~g) & \nabla \cdot s \geq 0 \end{matrix}

(d) Calculate the law of local energy conservation,

u \cdot \nabla \cdot T = 0

, for a viscous fluid with heat flow. Combine with the first law of thermodynamics and with the law of baryon conservation to obtain

\begin{matrix} (22.16h) & T \nabla \cdot s = ζ θ^{2} + 2 η σ_{α β} σ^{α β} - q^{α} (T_{, α} / T + a_{α}) \end{matrix}

Interpret each term of this equation as a contribution to entropy generation (example:

2 η σ_{α β} σ^{α β}

describes entropy generation by viscous heating). [Note: The term

q^{α} a_{α}

is relativistic in origin. It is associated with the inertia of the flowing heat.]
(e) When one takes account of the inertia of the flowing heat, one obtains the following generalization of the classical law of heat conduction:

\begin{matrix} (22.16i) & q^{α} = - κ P^{α β} (T_{, β} + T a_{β}) \end{matrix}

(Eckart 1940). Here

κ

is the coefficient of thermal conductivity. Use this equation to show that, for a fluid at rest in a stationary gravitational field (Exercise 22.5),
$$

\begin{matrix} (22.16j) & q_{0} = 0, q_{j} = - \frac{κ}{\sqrt{- g_{00}}} {(T \sqrt{- g_{00}})}_{, j} \end{matrix}

[T h u s, t h e r m a l e q u i l i b r i u m c o r r e s p o n d s n o t t o c o n s t a n t t e m p e r a t u r e, b u t t o t h e r e d s h i f t e d t e m p e r a t u r e d i s t r i b u t i o n $ T \sqrt{- g_{00}} = $ c o n s t a n t; T o l m a n (1934 a), p . 313.] A l s o, u s e t h e i d e a l i z e d l a w o f h e a t c o n d u c t i o n (22.16 i) t o r e e x p r e s s t h e r a t e o f e n t r o p y g e n e r a t i o n a s

\begin{aligned} (22.16k) & T \nabla \cdot s & = ζ θ^{2} + 2 η σ_{α β} σ^{α β} + (κ / T) P^{α β} (T_{, α} + T a_{α}) (T_{, β} + T a_{β}) \\ \geq 0 \end{aligned}

$$
[For further details about heat flow and for discussions of the limitations of the above idealized description, see e.g.,

§ 4.18

of Ehlers (1971); also Marle (1969), Anderson (1970), Stewart (1971), and papers cited therein.]

Electric and magnetic fields

Maxwell equations and Lorentz force law

§22.4. ELECTRODYNAMICS IN CURVED SPACETIME

In a local Lorentz frame in the presence of gravity, an observer can measure the electric and magnetic fields

E

and

B

using the usual Lorentz force law for charged particles. As in special relativity, he can regard

E

and

B

as components of an electromagnetic field tensor,

F^{\hat{0} \hat{j}} = - F^{\hat{j \hat{0}} = E^{\hat{j}}, F^{\hat{j \hat{k}}} = ϵ^{\hat{j} \hat{ℓ}} B^{\hat{ı}};, .}

he can regard the charge and current densities as components of a 4-vector

J^{\hat{α}}

, and he can write Maxwell's equations and the Lorentz force equation in the special relativistic form,

\begin{matrix} F^{\hat{α} \hat{β}}_{, \hat{β}} = 4 π J^{\hat{α}}, F_{\hat{α} \hat{β}, \hat{γ}} + F_{\hat{β} \hat{γ}, \hat{α}} + F_{\hat{γ} \hat{α}, \hat{β}} = 0, \\ m a^{\hat{α}} = F^{\hat{α} \hat{β}} q u_{\hat{β}} (\binom{m = mass of particle, q = charge}{u^{\hat{α}} = 4 -velocity, a^{\hat{α}} = 4 -acceleration}) . \end{matrix}

In any other frame these equations will have the same form, but with commas replaced by semicolons

\begin{matrix} (22.17a) & F_{; β}^{α β} = 4 π J^{α} \\ (22.17b) & F_{α β; γ} + F_{β γ; α} + F_{γ α; β} = 0, \\ (22.17c) & m a^{α} = F^{α β} q u_{β} . \end{matrix}

These are the basic equations of electrodynamics in the presence of gravity. From them follows everything else. For example, as in special relativity, so also here (exercise 22.9), they imply the equation of charge conservation

\begin{matrix} (22.18a) & J_{; α}^{α} = 0 \end{matrix}

and for an electromagnetic field interacting with charged matter (exercise 22.10) they imply vanishing divergence for the sum of the stress-energy tensors

\begin{matrix} (22.18b) & {(T^{(EM) α β} + T^{(MATTER) α β})}_{; β} = 0 \end{matrix}

As in special relativity, so also here, one can introduce a vector potential

A^{μ}

. Replacing commas by semicolons in the usual special-relativistic expression for

F^{μ ν}

in terms of

A^{μ}

, one obtains

\begin{matrix} (22.19a) & F_{μ ν} = A_{ν; μ} - A_{μ; ν} \end{matrix}

If all is well, this equation should guarantee (as in special relativity) that the Maxwell equations (22.17b) are satisfied. Indeed, it does, as one sees in exercise 22.8. To derive the wave equation that governs the vector potential, insert expression (22.19a) into the remaining Maxwell equations (22.17a), obtaining

\begin{matrix} (22.19b) & - A^{α; β}_{β} + A^{β; α}_{β} = 4 π J^{α} \end{matrix}

then commute covariant derivatives in the first term using the identity ( 16.6 c ), to obtain

\begin{matrix} (') & - A^{α; μ}_{μ} + A^{μ}_{; μ}^{; α} + R^{α}_{μ} A^{μ} = 4 π J^{α} . \end{matrix}

Finally, adopting the standard approach of special relativity, impose the Lorentz gauge condition

\begin{matrix} (22.19c) & A^{μ}_{; μ} = 0, \end{matrix}

thereby bringing the wave equation (

22.19 b^{'}

) into the form

\begin{matrix} (22.19d) & {(Δ_{d R} A)}^{α} \equiv - A^{α; β}_{β} + R^{α}_{β} A^{β} = 4 π J^{α} . \end{matrix}

The "de Rham vector wave operator"

Δ

which appears here is, apart from sign, a generalized d'Alambertian for vectors in curved spacetime. Mathematically it is more powerful than

- A^{α; β}_{; β}

, and than any other operator that reduces to (minus) the d'Alambertian in special relativity. [For a discussion, see de Rham (1955).]
Although the electrodynamic equations (22.17a)-(22.19b) are all obtained from special relativity by the comma-goes-to-semicolon rule, the wave equation ( 22.19 d ) for the vector potential is not ("curvature coupling"; see Box 16.1). Nevertheless, when spacetime is flat (so

R^{α}_{β} = 0

), ( 22.19 d ) does reduce to the usual wave equation of special relativity.

Exercise 22.8. THE VECTOR POTENTIAL FOR ELECTRODYNAMICS

Show that in any coordinate frame the connection coefficients cancel out of both equations (22.19a) and (22.17b), so they can be written

\begin{matrix} (22.20a) & F_{μ ν} = A_{ν, μ} - A_{μ, v}, \\ (22.20b) & F_{α β, γ} + F_{β γ, α} + F_{γ α, β} = 0 . \end{matrix}

(In the language of differential forms these equations are

F = d A, d F = 0

.) Then use this form of the equations to show that equation (22.19a) implies equation (22.17b), as asserted in the text.

Exercise 22.9. CHARGE CONSERVATION IN THE PRESENCE OF GRAVITY

Show that Maxwell's equations (22.17a,b) imply the equation of charge conservation (22.18a) when gravity is present, just as they do in special relativity theory. [Hints: Use the antisymmetry of

F^{α β}

; and beware of the noncommutation of the covariant derivatives, which must be handled using equations (16.6). Alternatively, show that in coordinate frames, equation (22.17a) can be written as

\begin{matrix} (') & \frac{1}{\sqrt{| g |}} \frac{\partial}{\partial x^{β}} (\sqrt{| g |} F^{α β}) = 4 π J^{α} \end{matrix}

and (22.18a) as

\begin{matrix} (') & J_{; α}^{α} \equiv \frac{1}{\sqrt{| g |}} \frac{\partial}{\partial x^{α}} (\sqrt{| g |} J^{α}) = 0 \end{matrix}

and carry out the demonstration in a coordinate frame.]

Exercise 22.10. INTERACTING ELECTROMAGNETIC FIELD AND CHARGED MATTER

As in special relativity, so also in the presence of gravity ("equivalence principle"), the stress-energy tensor for an electromagnetic field is

\begin{matrix} (22.21) & T^{(EM)}_{α β} = \frac{1}{4 π} (F_{α μ} F_{β}^{μ} - \frac{1}{4} F_{μ ν} F^{μ ν} g_{α β}) . \end{matrix}

Use Maxwell's equations

(22.17 a, b)

in the presence of gravity to show that

\begin{matrix} (22.22) & T^{(EM) α β}_{: β} = - F^{α β} J_{β} . \end{matrix}

But

F^{α β} J_{β}

is just the Lorentz 4-force per unit volume with which the electromagnetic field acts on the charged matter [see the Lorentz force equation (22.17c); also equation (5.43)]; i.e., it is

T^{(MATTER) α β}_{; β}

. Consequently, the above equation can be rewritten in the form (22.18b) cited in the text.

§22.5. GEOMETRIC OPTICS IN CURVED SPACETIME*

Radio waves from the quasar 3 C 279 pass near the sun and get deflected by its gravitational field. Light rays emitted by newborn galaxies long ago and far away propagate through the cosmologically curved spacetime of the universe, and get focused (and redshifted) producing curvature-enlarged (but dim) images of the galaxies on the Earth's sky.

These and most other instances of the propagation of light and radio waves are subject to the laws of geometric optics. This section derives those laws, in curved spacetime, from Maxwell's equations.

The fundamental laws of geometric optics are: (1) light rays are null geodesics; (2) the polarization vector is perpendicular to the rays and is parallel-propagated along the rays; and (3) the amplitude is governed by an adiabatic invariant which, in quantum language, states that the number of photons is conserved.

The conditions under which these laws hold are defined by conditions on three lengths: (1) the typical reduced wavelength of the waves,

\begin{matrix} (22.23a) & λ \equiv \frac{λ}{2 π} = (\binom{"classical distance of closest approach for}{a photon with one unit of angular momentum""}), \end{matrix}

as measured in a typical local Lorentz frame (e.g., a frame at rest relative to nearby galaxies); (2) the typical length

E

over which the amplitude, polarization, and wavelength of the waves vary, e.g., the radius of curvature of a wave front, or the length of a wave packet produced by a sudden outburst in a quasar; (3) the typical radius of curvature

R

of the spacetime through which the waves propagate,

\begin{matrix} (22.23b) & R \equiv {| \begin{array}{l} typical component of Riemann as measured \\ in typical local Lorentz frame \end{array} |}^{- 1 / 2} \end{matrix}

Geometric optics is valid whenever the reduced wavelength is very short compared to each of the other scales present,

\begin{matrix} (22.23c) & λ ≪ E and λ ≪ Ω, \end{matrix}

so that the waves can be regarded locally as plane waves propagating through spacetime of negligible curvature.
Mathematically one exploits the geometric-optics assumption,

λ ≪ E

and

λ ≪ R

, as follows. Focus attention on waves that are highly monochromatic over regions

\leq E

. (More complex spectra can be analyzed by superposition, i.e., by Fourier analysis.) Split the vector potential of electromagnetic theory into a rapidly changing, real phase,

θ \sim (distance propagated) / π

and a slowly changing, complex amplitude (i.e. one with real and imaginary parts),

A = Real part of {amplitude \times e^{i θ}} \equiv X {amplitude \times e^{i θ}} .

Imagine holding fixed the scale of the amplitude variation,

L

, and the scale of the spacetime curvature,

R

, while making the reduced wavelength,

λ

, shorter and shorter. The phase will get larger and larger

(θ \propto 1 / λ)

at any fixed event in spacetime, but the amplitude as a function of location in spacetime can remain virtually unchanged,

Amplitude

= [\begin{array}{l} dominant part, \\ independent of λ \end{array}] + [\begin{array}{l} small corrections (deviations from \\ geometric optics) due to finite wavelength \end{array}]

Overview of geometric optics

Conditions for validity of geometric optics

The "two-length-scale" expansion underlying geometric optics

This circumstance allows one to expand the amplitude in powers of

λ

\begin{matrix} Amplitude = \underset{↑}{a} + \underset{c}{b} + \underset{↑}{c} + \dots . \\ [\begin{matrix} independent \\ of t \end{matrix}] \end{matrix}

[Actually, the expansion proceeds in powers of the dimensionless number

\begin{matrix} (22.24) & λ / (minimum of E and R) \equiv λ / L \end{matrix}

Applied mathematicians call this a "two-length-scale expansion"; see, e.g., Cole (1968). The basic short-wavelength approximation here has a long history; see, e.g., Liouville (1837), Rayleigh (1912). Following a suggestion of Debye, it was applied to Maxwell's equations by Sommerfeld and Runge (1911). It is familiar as the WKB approximation in quantum mechanics, and has many other applications as indicated by the bibliography in Keller, Lewis, and Seckler (1956). The contribution of higher order terms is considered by Kline (1954) and Lewis (1958). See especially the book of Fröman and Fröman (1965).]
It is useful to introduce a parameter

ε

that keeps track of how rapidly various terms approach zero (or infinity) as

λ / L

approaches zero:

\begin{matrix} (22.25) & A_{μ} = x {(a_{μ} + ε b_{μ} + ε^{2} c_{μ} + \dots) e^{i θ / ε}} . \end{matrix}

Any term with a factor

ε^{n}

in front of it varies as

(λ / L)^{n}

in the limit of very small wavelengths

[θ \propto (λ / L)^{- 1}; c_{μ} \propto (λ / L)^{2}

; etc.]. By convention,

ε

is a dummy expansion parameter with eventual value unity; so it can be dropped from the calculations when it ceases to be useful. And by convention, all "post-geometric-optics corrections" are put into the amplitude terms

b, c, \dots

; none are put into

θ

.
Note that, while the phase

θ

is a real function of position in spacetime, the amplitude and hence the vectors

a, b, c, \dots

are complex. For example, to describe monochromatic waves with righthand circular polarization, propagating in the

z

direction, one could set

θ = ω (z - t)

and

a = 1 / \sqrt{2} a (e_{x} + i e_{y})

with

a

real; so

A = ℵ {\frac{1}{\sqrt{2}} a (e_{x} + i e_{y}) e^{i ω (z - t)}} = \frac{1}{\sqrt{2}} a {\cos [ω (z - t)] e_{x} - \sin [ω (z - t)] e_{y}}

The assumed form (22.25) for the vector potential is the mathematical foundation of geometric optics. All the key equations of geometric optics result from inserting this vector potential into the source-free wave equation

Δ A = 0

[equation (22.19d)] and into the Lorentz gauge condition

\nabla \cdot A = 0

[equation (22.19c)]. The resulting equations (derived below) take their simplest form only when expressed in terms of the following:

Basic concepts of geometric
optics:

The vector potential in geometric optics

\begin{matrix} (22.26a) & "wave vector," k \equiv \nabla θ \end{matrix}

\begin{matrix} (22.26b) & "scalar amplitude," a \equiv (a \cdot \overset{―}{a})^{1 / 2} = {(a^{μ} {\bar{a}}_{μ})}^{1 / 2} \end{matrix}

"polarization vector,"

f \equiv a / a =

"unit complex vector along

a

". (22.26c)
(Here

\overset{―}{a}

is the complex conjugate of

a

.) Light rays are defined to be the curves

P (λ)

normal to surfaces of constant phase

θ

. Since

k \equiv \nabla θ

is the normal to these surfaces, the differential equation for a light ray is

\begin{matrix} (22.26~d) & \frac{d x^{μ}}{d λ} = k^{μ} (x) = g^{μ ν} (x) θ_{, v} (x) \end{matrix}

Box 22.3, appropriate for study at this point, shows the polarization vector, wave vector, surfaces of constant phase, and light rays for a propagating wave; the scalar amplitude, not shown there, merely tells the length of the vector amplitude

a

. Insight into the complex polarization vector, if not familiar from electrodynamics, can be developed later in Exercise 22.12.

So much for the foundations. Now for the calculations. First insert the geometricoptics vector potential (22.25) into the Lorentz gauge condition:

\begin{matrix} (22.27) & 0 = A_{; μ}^{μ} = x {[\frac{i}{ε} k_{μ} (a^{μ} + ε b^{μ} + \dots) + {(a^{μ} + ε b^{μ} + \dots)}_{; μ}] e^{i θ / ε}} . \end{matrix}

The leading term (order

1 / ε

) says

\begin{matrix} (22.28) & k \cdot a = 0 (amplitude is perpendicular to wave vector) \end{matrix}

or, equivalently

\begin{matrix} (') & k \cdot f = 0 (polarization is perpendicular to wave vector). \end{matrix}

The post-geometric-optics breakdown in this orthogonality condition is governed by the higher-order terms

[0 (1), 0 (ε), 0 (ε^{2}), \dots]

in the gauge condition (22.27); for example, the

0 (1)

terms say

k \cdot b = i \nabla \cdot a

Next insert the vector potential (22.25) into the source-free wave equation (22.19d):

\begin{aligned} 0 = & {(Δ_{d R} A)}^{α} = - A^{α; β}_{β} + R^{α}_{β} A^{β} \\ = & ℵ {[\frac{1}{ε^{2}} k^{β} k_{β} (a^{α} + ε b^{α} + ε^{2} c^{α} + \dots) - 2 \frac{i}{ε} k^{β} {(a^{α} + ε b^{α} + \dots)}_{; β} \\ (22.29) & - \frac{i}{ε} k^{β}_{; β} (a^{α} + ε b^{α} + \dots) - {(a^{α} + \dots)}^{; β}_{β} + R^{α}_{β} (a^{β} + \dots)] e^{i θ / ε}} . \end{aligned}

Collect terms of order

1 / ε^{2}

and

1 / ε

(terms of order higher than

1 / ε

govern post-geometric-optics corrections):

Box 22.3 GEOMETRY OF AN ELECTROMAGNETIC WAVE TRAIN

The drawing shows surfaces of constant phase,

θ =

constant, emerging through the "surface of simultaneity",

t = 0

, of a local Lorentz frame. The surfaces shown are alternately "crests"

(θ = 1764 π, θ = 1766 π, \dots)

and "troughs"

(θ = 1765 π, θ =

1767 π, \dots

) of the wave train. These surfaces make up a 1 -form,

\tilde{k} = d θ

. The "corresponding vector"

k = \nabla θ

is the "wave vector." The wave vector is null,

k \cdot k = 0

, according to Maxwell's equations [equation (22.30)]. Therefore it lies in a surface of constant phase:

(\binom{number of surfaces}{pierced by k}) = ⟨ d θ, k ⟩ = ⟨ \tilde{κ}, k ⟩ = k \cdot k = 0 .

But not only does it lie in a surface of constant phase; it is also perpendicular to that surface! Any vector

v

in that surface must satisfy

k \cdot v = ⟨ \tilde{k}, v ⟩ = ⟨ d θ, v ⟩ = 0

because it pierces no surfaces.

Geometric optics assumes that the reduced wavelength

λ

, as measured in a typical local Lorentz frame, is small compared to the scale

L

of inhomogeneities in the wave train and small compared to the radius of curvature of spacetime,

R

. Thus, over regions much larger than

λ

but smaller than

E

R

, the waves are plane-fronted
and monochromatic, and there exist Lorentz reference frames (Riemann normal coordinates). In one of these "extended" local Lorentz frames, the phase must be

θ = k_{α} x^{α} + constant;

no other expression will yield

\nabla θ = k

. The corresponding vector potential [equation (22.25)] will be

A^{μ} = ℵ {a^{μ} \exp [i (k \cdot x - k^{0} t)]} + ("post-geometric-optics corrections");

hence,

\begin{matrix} k^{0} = 2 π / (period of wave) = 2 π ν = ω \equiv (angular frequency), \\ | k | = 2 π / (wavelength of wave) = 1 / λ = ω \end{matrix}

k

points along direction of propagation of wave.
At each event in spacetime there is a wave vector; and these wave vectors, tacked end-on-end, form a family of curves-the "light rays" or simply "rays"-whose tangent vector is

k

. The rays, like their tangent vector, lie both in and perpendicular to the surfaces of constant phase.

The affine parameter

λ

of a ray (not to be confused with wavelength

= 2 π λ

) satisfies

k = d / d λ

; therefore it is given by

λ = t / k^{0} + constant = t / ω + constant

where

t

is proper time along the ray as measured, not by the ray itself (its proper time is zero!), but by the local Lorentz observer who sees angular frequency

ω

. Thus, while

ω

is a frame-dependent quantity and

t

is also a frame-dependent quantity, their quotient

t / ω

when measured along the ray (not off the ray) is the frame-independent affine parameter. For a particle it is possible and natural to identify the affine parameter

λ

with proper time

τ

. For a light ray this identification is unnatural and impossible. The lapse of proper time along the ray is identically zero. The springing up of

λ

to take the place of the vanished

τ

gives one a tool to do what one might not have suspected to be possible. Given a light ray shot out at event

a

and passing through event

B

, one can give a third event

C

along the same null world line that is twice as "far" from

a

B

is "far," in a new sense of "far" that has nothing whatever directly to do with proper time (zero!), but is defined by equal increments of the affine parameter

(λ_{e} - λ_{g h} = λ_{g h} - λ_{d})

. The "affine parameter" has a meaning for any null geodesic analyzed even in isolation. In this respect, it is to be distinguished from the so-called "luminosity distance" which is sometimes introduced in dealing with the propagation of radiation through curved spacetime, and which is defined by the spreading apart of two or more light rays coming from a common source.

Maxwell's equations as explored in the text [equation (22.28')] guarantee that the complex polarization vector

f

is perpendicular to the wave vector

k

and that, therefore, it lies in a surface of constant phase (see drawing). Intuition into the polarization vector is developed in exercise 22.12.

\begin{aligned} 0 (\frac{1}{ε^{2}}) : k^{β} k_{β} a^{α} = 0 \\ (22.30) & ⟹ k \cdot k = 0 (wave vector is null); \\ 0 (\frac{1}{ε}) : \underset{ζ = 0]}{\underset{⏟}{k^{β} k_{β} b^{α} - 2 i (k^{β} a^{α}_{; β} + \frac{1}{2} k^{β}_{; β} a^{α}) = 0}} \\ (22.31) & ⟹ \nabla_{k} a = - \frac{1}{2} (\nabla \cdot k) a (propagation equation for vector amplitude). \end{aligned}

These equations

(22.30, 22.31)

together with equation (22.28) are the basis from which all subsequent results will follow. As a first consequence, one can obtain the geodesic law from equation (22.30). Form the gradient of

k \cdot k = 0

0 = {(k^{β} k_{β})}_{; α} = 2 k^{β} k_{β; α}

and use the fact that

k_{β} \equiv θ_{, β}

is the gradient of a scalar to interchange indices,

θ_{; β α} = θ_{; α β}

0 = k^{β} k_{β; α} = k^{β} k_{α; β} .

The main laws of geometric optics:

The result is

\begin{matrix} (22.32) & \nabla_{k} k = 0 (propagation equation for wave vector). \end{matrix}

Notice that this is the geodesic equation! Combined with equation (22.30), it is the statement, derived from Maxwell's equations in curved spacetime, that light rays are null geodesics, the first main result of geometric optics.
Turn now from the propagation vector

k = \nabla θ

to the wave amplitude

a = a f

, and obtain separate equations for the magnitude

a

and polarization

f

. Use equation (22.31) to compute

\begin{aligned} 2 a \partial_{k} a & = 2 a \nabla_{k} a = \nabla_{k} a^{2} = \nabla_{k} (a \cdot \overset{―}{a}) = \overset{―}{a} \cdot \nabla_{k} a + a \cdot \nabla_{k} \overset{―}{a} \\ = - \frac{1}{2} (\nabla \cdot k) (\overset{―}{a} \cdot a + a \cdot \overset{―}{a}) = - a^{2} \nabla \cdot k; \end{aligned}

\begin{matrix} (22.33) & \partial_{k} a = - \frac{1}{2} (\nabla \cdot k) a (propagation equation for scalar amplitude) . \end{matrix}

Next write

a = a f

in equation (22.31) to obtain

0 = \nabla_{k} (a f) + \frac{1}{2} (\nabla \cdot k) a f = a \nabla_{k} f + f [\nabla_{k} a + \frac{1}{2} (\nabla \cdot k) a] = a \nabla_{k} f

\begin{matrix} (22.34) & \nabla_{k} f = 0 (propagation equation for polarization vector). \end{matrix}

This together with equation (22.28'), constitutes the second main result of geometric optics, that the polarization vector is perpendicular to the rays and is parallel-propagated along the rays. It is now possible to see that these results, derived from equations (22.30) and (22.31) are consistent with the gauge condition (22.28). The vectors

k

and

f

, specified at one point, are fixed along the entire ray by their propagation equations. But because both propagation equations are parallel-transport laws, the conditions

k \cdot k = 0, f \cdot \overset{―}{f} = 1

, and

k \cdot f = 0

, once imposed on the vectors at one point, will be satisfied along the entire ray.

The equation (22.33) for the scalar amplitude can be reformulated as a conservation law. Since

\partial_{k} \equiv (k \cdot \nabla)

, one rewrites the equation as

(k \cdot \nabla) a^{2} + a^{2} \nabla \cdot k = 0

, or

\begin{matrix} (22.35) & \nabla \cdot (a^{2} k) = 0 \end{matrix}

Consequently the vector

a^{2} k

is a "conserved current," and the integral

\int a^{2} k^{μ} d^{3} Σ_{μ}

has a fixed, unchanging value for each 3 -volume cutting a given tube formed of light rays. (The tube must be so formed of rays that an integral of

a^{2} k

over the walls of the tube will give zero.) What is conserved? To remain purely classical, one could say it is the "number of light rays" and call

a^{2} k^{0}

the "density of light rays" on an

x^{0} =

constant hypersurface. But the proper correspondence and more concrete physical interpretation make one prefer to call equation (22.35) the law of conservation of photon number. It is the third main result of geometric optics. Photon number, of course, is not always conserved; it is an adiabatic invariant, a quantity that is not changed by influences (e.g., spacetime curvature,

\sim 1 / R^{2}

) which change slowly

(ℜ ≫ λ)

compared to the photon frequency.

Box 22.4 summarizes the above equations of geometric optics, along with others derived in the exercises.
(2) polarization vector is perpendicular to ray and is parallel propagated along ray
(3) conservation of "photon number"

Exercise 22.11. ELECTROMAGNETIC FIELD AND STRESS ENERGY

Derive the equations given in part D of Box 22.4 for

F, E, B

, and

T

Exercise 22.12. POLARIZATION

At an event

P_{0}

through which geometric-optics waves are passing, introduce a local Lorentz frame with

z

-axis along the direction of propagation. Then

k = ω (e_{0} + e_{z})

. Since the polarization vector is orthogonal to

k

, it is

f = f^{0} (e_{0} + e_{z}) + f^{1} e_{x} + f^{2} e_{y}

; and since

f \cdot \overset{―}{f} = 1

, it has

{| f^{1} |}^{2} + {| f^{2} |}^{2} = 1

.
(a) Show that the component

f^{0}

of the polarization vector has no influence on the electric and magnetic fields measured in the given frame; i.e., show that one can add a multiple of

k

f

without affecting any physical measurements.
(continued on page 581)

EXERCISES

Box 22.4 GEOMETRIC OPTICS IN CURVED SPACETIME (Summary of Results Derived in Text and Exercises)

A. Geometric Optics Assumption

Electromagnetic waves propagating in a source-free region of spacetime are locally plane-fronted and monochromatic (reduced wavelength

λ ≪

scale

E

over which amplitude, wavelength, or polarization vary; and

λ ≪ R =

mean radius of curvature of spacetime).

B. Rays, Phase, and Wave Vector (see Box 22.3)

Everything (amplitude, polarization, energy, etc.) is transported along rays; and the quantities on one ray do not influence the quantities on any other ray.
The rays are null geodesics of curved spacetime, with tangent vectors ("wave vectors")

k

\nabla_{k} k = 0

The rays both lie in and are perpendicular to surfaces of constant phase,

θ =

const.; and their tangent vectors are the gradient of

θ

k = \nabla θ .

In a local Lorentz frame,

k^{0}

is the "angular frequency" and

k^{0} / 2 π

is the ordinary frequency of the waves, and

n = k / k^{0}

is a unit 3 -vector pointing along their direction of propagation.

C. Amplitude and Polarization Vector

The waves are characterized by a real amplitude

a

and a complex polarization vector

f

of unit length,

f \cdot \overset{―}{f} = 1

. (Of the fundamental quantities

θ, k, a, f

, all are real except

f

. See exercise 22.12 for deeper understanding of

f

.)
The polarization vector is everywhere orthogonal to the rays,

k \cdot f = 0

; and is parallel-transported along them,

\nabla_{k} f = 0

.
The propagation law for the amplitude is

\partial_{k} a = - \frac{1}{2} (\nabla \cdot K) a .

This propagation law is equivalent to a law of conservation of photons (classically: of rays);

a^{2} k

is the "conserved current" satisfying

\nabla \cdot (a^{2} k) = 0

; and

(8 π ℏ)^{- 1} \int a^{2} k^{0} \sqrt{| g |} d^{3} x

is the number of photons (rays) in the 3 -volume of integration on any

x^{0} =

constant hypersurface, and is constant as this volume is carried along the rays.
The propagation law holds separately on each hypersurface of constant phase. There it can be interpreted as conservation of a

a^{2} a

, where

a

is a two-dimensional cross-sectional area of a pulse of photons or rays. See exercise 22.13.

D. Vector Potential, Electromagnetic Field, and Stress-Energy-Momentum

At any event the vector potential in Lorentz gauge is

A = x {a e^{i θ} f},

where

k_{i}

denotes the real part.
The electromagnetic field tensor is orthogonal to the rays,

F \cdot k = 0

, and is given by

F = x {i a e^{i θ} k \land f} .

The corresponding electric and magnetic fields in any local Lorentz frame are

\begin{matrix} E = i {i a k^{0} e^{i θ} (projection of f perpendicular to k)}, \\ B = n \times E, where n \equiv k / k^{0} . \end{matrix}

The stress-energy tensor, averaged over a wavelength, is

T = (1 / 8 π) a^{2} k \otimes k

corresponding to an energy density in a local Lorentz frame of

T^{00} = (1 / 8 π) {(a k^{0})}^{2}

and an energy flux of

T^{0 j} = T^{00} n^{j}

so that energy flows along the rays (in

n = k / k^{0}

direction) with the speed of light. This is identical with the stress-energy tensor that would be produced by a beam of photons with 4-momenta

p = ℏ k

.
Conservation of energy-momentum

\nabla \cdot T = 0

follows from the ray conservation law

\nabla \cdot (a^{2} k) = 0

and the geodesic law

\nabla_{k} k \equiv (k \cdot \nabla) k = 0

8 π \nabla \cdot T = \nabla \cdot (a^{2} k \otimes k) = [\nabla \cdot (a^{2} k)] k + a^{2} (k \cdot \nabla) k = 0 .

Box 22.4 (continued)

The adiabatic (geometric optics) invariant "ray number"

a^{2} k^{0}

or "photon number"

(8 π ℏ)^{- 1} a^{2} k^{0}

in a unit volume is proportional to the energy,

(8 π)^{- 1} a^{2} {(k^{0})}^{2}

, divided by the frequency,

k^{0}

-corresponding exactly to the harmonic oscillator adiabatic invariant

E / ω

[Einstein (1912), Ehrenfest (1916), Landau and Lifshitz (1960)].

E. Photon Reinterpretation of Geometric Optics

The laws of geometric optics can be reinterpreted as follows. This reinterpretation becomes a foundation of the standard quantum theory of the electromagnetic field (see, e.g., Chapters 1 and 13 of Baym (1969)]; and the classical limit of that quantum theory is standard Maxwell electrodynamics.
Photons are particles of zero rest mass that move along null geodesics of spacetime (the null rays).
The 4-momentum of a photon is related to the tangent vector of the null ray (wave vector) by

p = ℏ k

. A renormalization of the affine parameter,

(new parameter) = (1 / ℏ) \times (old parameter),

makes

p

the tangent vector to the ray.
Each photon possesses a polarization vector,

f

, which is orthogonal to its 4 -momentum

(p \cdot f = 0)

, and which it parallel-transports along its geodesic world line

(\nabla_{p} f = 0)

.
A swarm of photons, all with nearly the same 4-momentum

p

and polarization vector

f

(as compared by parallel transport), make up a classical electromagnetic wave. The scalar amplitude

a

of the wave is determined by equating the stress-energy tensor of the wave

T = \frac{1}{8 π} a^{2} k \otimes k = \frac{1}{8 π} {(\frac{a}{ℏ})}^{2} p \otimes p

to the stress-energy tensor of a swarm of photons with number-flux vector

S

T = p \otimes S

[see equation (5.18)]. The result:

S = \frac{1}{8 π} {(\frac{a}{ℏ})}^{2} p = \frac{1}{8 π ℏ} a^{2} k

or, in any local Lorentz frame,

a = {(8 π ℏ^{2} S^{0} / p^{0})}^{1 / 2} = (8 π)^{1 / 2} ℏ {(\frac{number density of photons}{energy of one photon})}^{1 / 2} .

(b) Show that the following polarization vectors correspond to the types of polarization listed:

\begin{aligned} f = e_{x}, linear polarization in x direction; \\ f = e_{y}, linear polarization in y direction; \\ f = \frac{1}{\sqrt{2}} (e_{x} + i e_{y}), righthand circular polarization; \\ f = \frac{1}{\sqrt{2}} (e_{x} - i e_{y}), lefthand circular polarization; \\ f = α e_{x} + i {(1 - α^{2})}^{1 / 2} e_{y}, righthand elliptical polarization. \end{aligned}

(c) Show that the type of polarization (linear; circular; elliptical with given eccentricity of ellipse) is the same as viewed in any local Lorentz frame at any event along a given ray. [Hint: Use pictures and abstract calculations rather than Lorentz transformations and component calculations.]

Exercise 22.13. THE AREA OF A bundle of rays

Write equation (22.31) in a coordinate system in which one of the coordinates is chosen to be

x^{0} = θ

, the phase (a retarded time coordinate).
(a) Show that

g^{00} = 0

and that no derivatives

\partial / \partial θ

appear in equation (22.33); so propagation of

a

can be described within a single

θ =

constant hypersurface.
(b) Perform the following construction (see Figure 22.1). Pick a ray

C_{0}

along which

a

is to be propagated. Pick a bundle of rays, with two-dimensional cross section, that (i) all lie in the same constant-phase surface as

E_{0}

, and (ii) surround

E_{0}

. (The surface is three-di-

Figure 22.1.
Geometric optics for a bundle of rays with two-dimensional cross section, all lying in a surface of constant phase,

θ =

const. Sketch (a) shows the bundle, surrounding a central ray

e_{0}

, in a spacetime diagram with one spatial dimension suppressed. Sketch (b) shows the bundle as viewed on a slice of simultaneity in a local Lorentz frame at the event

Φ_{0}

. Slicing the bundle turns each ray into a "photon"; so the bundle becomes a two-dimensional surface filled with photons. The area

Ω

of this photon-filled surface obeys the following laws (see exercises 22.13 and 22.14): (1)

C

is independent of the choice of Lorentz frame; it depends only on location

P_{0}

along the ray

C_{0}

. (2) The amplitude

a

of the waves satisfies

a a^{2} = constant all along the ray e_{0}

("conservation of photon flux"). (3) a obeys the "propagation equation" (22.36).
mensional, so any bundle filling it has a two-dimensional cross section.) At any event

P_{0}

, in any local Lorentz frame there, on a "slice of simultaneity"

x^{0} =

constant, measure the cross-sectional area

a

of the bundle. (Note: the area being measured is perpendicular to

k

in the three-dimensional Euclidean sense; it can be thought of as the region occupied momentarily by a group of photons propagating along, side by side, in the

k

direction.) Show that the area

C

is the same, at a given event

P_{0}

, regardless of what Lorentz frame is used to measure it; but the area changes from point to point along the ray

C_{0}

as a result of the rays' divergence away from each other or convergence toward each other:

\begin{matrix} (22.36) & \partial_{k} a = (\nabla \cdot k) a . \end{matrix}

Then show that

C a a^{2}

is a constant everywhere along the ray

C_{0}

("conservation of photon flux"). [Hints: (i) Any vector

ξ

connecting adjacent rays in the bundle is perpendicular to

k

, because

ξ

lies in a surface of constant

θ

and

k \cdot ξ = ⟨ \tilde{κ}, ξ ⟩ = ⟨ d θ, ξ ⟩ =

(change in

θ

along

ξ

)

= 0

. (ii) Consider, for simplicity, a bundle with rectangular cross section as seen in a specific local Lorentz frame at a specific event

P_{0}

[edge vectors

v

and

w

with

v \cdot w = 0

(edges perpendicular) and

v \cdot e_{0} = w \cdot e_{0} = 0

(edges in surface of constant time) and

v \cdot k = w \cdot k = 0

(since edge vectors connect adjacent rays of the bundle)]. Show pictorially that in any other Lorentz frame at

P_{0}

, the edge vectors are

v^{'} = v + α k

and

w^{'} = w + β k

for some

α

and

β

. Conclude that in all Lorentz frames at

P_{0}

the cross section has identical shape and identical area, and is spatially perpendicular to the direction of propagation

(k \cdot v = k \cdot w = 0

). (iii) By a calculation in a local Lorentz frame show that

\partial_{k} d = (\nabla \cdot k) d

. (iv) Conclude from

\partial_{k} a = - \frac{1}{2} (\nabla \cdot k) a

that

\partial_{k} (a a^{2}) = 0

Exercise 22.14. FOCUSING THEOREM

The cross-sectional area

a

of a bundle of rays all lying in the same surface of constant phase changes along the central ray of the bundle at the rate (22.36) (see Figure 22.1).
(a) Derive the following equation ("focusing equation") for the second derivative of

a^{1 / 2}

\begin{matrix} (22.37) & \frac{d^{2} Q^{1 / 2}}{d λ^{2}} = - (| σ |^{2} + \frac{1}{2} R_{α β} k^{α} k^{β}) a^{1 / 2} \end{matrix}

where

λ

is affine parameter along the central ray

(k = d / d λ)

, and the "magnitude of the shear of the rays",

| σ |

, is defined by the equation

\begin{matrix} (22.38) & | σ |^{2} \equiv \frac{1}{2} k_{α; β} k^{α; β} - \frac{1}{4} {(k_{; μ}^{μ})}^{2} \end{matrix}

[Hint: This is a vigorous exercise in index manipulations. The key equations needed in the manipulations are

a_{, α} k^{α} = (k^{α}_{; α}) Q

[equation (22.36)];

k^{α}_{; β} k^{β} = 0

[geodesic equation (22.32) for rays];

k_{α; β} = k_{β; α}

[which follows from

k_{α} \equiv θ_{, α}]

; and the rule (16.6c) for interchanging covariant derivatives of a vector.]
(b) Show that, in a local Lorentz frame where

k = ω (e_{t} + e_{z})

at the origin,

\begin{matrix} (22.39) & | σ |^{2} = \frac{1}{4} {(k_{x, x} - k_{y, y})}^{2} + {(k_{x, y})}^{2} . \end{matrix}

Thus,

| σ |^{2}

is nonnegative, which justifies the use of the absolute value sign.
(c) Discussion: The quantity

| σ |

is called the shear of the bundle of rays because it measures the extent to which neighboring rays are sliding past each other [see, e.g., Sachs (1964)]. Hence, the focusing equation (22.37) says that shear focuses a bundle of rays (makes

d^{2} a^{1 / 2} / d λ^{2} < 0

); and spacetime curvature also focuses it if

R_{α β} k^{α} k^{β} > 0

, but defocuses it if

R_{α β} k^{α} k^{β} < 0

. (When a bundle of toothpicks, originally circular in cross section, is squeezed into an elliptic cross section, it is sheared.)
(d) Assume that the energy density

T_{\hat{0} \hat{0}}

, as measured by any observer anywhere in spacetime, is nonnegative. By combining the focusing equation (22.37) with the Einstein field equation, conclude that

\begin{matrix} (22.40) & \frac{d^{2} a^{1 / 2}}{d λ^{2}} \leq 0 (\begin{array}{l} for any bundle of rays, all in the same \\ surface of constant phase, anywhere in \\ spacetime \end{array}) \end{matrix}

(focusing theorem). This theorem plays a crucial role in black-hole physics (

§ 34.5

) and in the theory of singularities (

§ 34.6

§22.6. KINETIC THEORY IN CURVED SPACETIME*

The stars in a galaxy wander through spacetime, each on its own geodesic world line, each helping to produce the spacetime curvature felt by all the others. Photons, left over from the hot phases of the big bang, bathe the Earth, bringing with themselves data on the homogeneity and isotropy of the universe. Theoretical analyses of these and many other problems are unmanageable, if they attempt to keep track of the motion of every single star or photon. But a statistical description gives accurate results and is powerful. Moreover, for most problems in astrophysics and cosmology, the simplest of statistical descriptions-one ignoring collisions-is adequate. Usually collisions are unimportant for the large-scale behavior of a system (e.g.; a galaxy), or they are so important that a fluid description is possible (e.g., in a stellar interior).

Consider, then, a swarm of particles (stars, or photons, or black holes, or . . .) that move through spacetime on geodesic world lines, without colliding. Assume, for simplicity, that the particles all have the same rest mass. Then all information of a statistical nature about the particles can be incorporated into a single function, the "distribution function" or "number density in phase space",

O

Define

R

in terms of measurements made by a specific local Lorentz observer at a specific event

P_{0}

in curved spacetime. Give the observer a box with 3 -volume

V_{x}

(and with imaginary walls). Ask the observer to count how many particles,

N

, are inside the box and have local-Lorentz momentum components

p^{j}

in the range

P^{j} - \frac{1}{2} Δ p^{j} < p^{j} < P^{j} + \frac{1}{2} Δ p^{j}

(He can ignore the particle energies

p^{0}

; since all particles have the same rest mass

m

, energy

p^{0} = {(m^{2} + p^{2})}^{1 / 2}

Volume in phase space for a group of identical particles

Lorentz invariance of volume in phase space

Liouville's theorem (conservation of volume in phase space)

Number density in phase space (distribution function)
is fixed uniquely by momentum.) The volume in momentum space occupied by the

N

particles is

V_{p} = Δ p^{x} Δ p^{y} Δ p^{z}

; and the volume in phase space is

\begin{matrix} (22.41) & V \equiv V_{x} V_{p} \end{matrix}

Other observers at

P_{0}

, moving relative to the first, will disagree on how much spatial volume

V_{x}

and how much momentum volume

V_{p}

these same

N

particles occupy:

\begin{matrix} (22.42) & V_{x} and V_{p} depend on the choice of Lorentz fram.e. \end{matrix}

However, all observers will agree on the value of the product

V \equiv V_{x} V_{p}

("volume in phase space"):

The phase-space volume

V

occupied by a given set of

N

identical particles at a given event in spacetime is independent of the local Lorentz frame in which it is measured.
(See Box 22.5 for proof.) Moreover, as the same

N

particles move through spacetime along their geodesic world lines (and through momentum space), the volume

V

they span in phase space remains constant:

The

V

occupied by a given swarm of

N

particles is independent of location along the world line of the swarm ("Liouville's theorem in curved spacetime").
(See Box 22.6 for proof.)
More convenient for applications than the volume

V

in phase space occupied by a given set of

N

particles is the "number density in phase space" ("distribution function") in the neighborhood of one of these particles:

\begin{matrix} (22.45) & R \equiv N / V \end{matrix}

On what does this number density depend? It depends on the location in spacetime,

P

, at which the measurements are made. It also depends on the 4-momentum

p

of the particle in whose neighborhood the measurements are made. But because the particles all have the same rest mass,

p

cannot take on any and every value in the tangent space at

P

. Rather,

p

is confined to the "forward mass hyperboloid" at

P

p^{2} = m^{2}; p lies inside future light cone.

Thus,

\begin{matrix} (22.46) & R = R [(\binom{location, P,}{in spacetime}), (\begin{array}{l} 4 -momentum p, which must lie \\ on the forward mass hyperboloid \\ of the tangent space at P \end{array})] . \end{matrix}

Pick some one particle in the swarm, with geodesic world line

P (λ) [λ =

(affine parameter

) = (

proper time, if particle has finite rest mass)], and with 4-momentum

Box 22.5 VOLUME IN PHASE SPACE

A. For Swarm of Identical Particles with Nonzero Rest Mass

Pick an event

P_{0}

, through which passes a particle named "John" with a 4 -momentum named "

P

". In John's local Lorentz rest frame at

P_{0}

("barred frame",

\bar{s}

), select a small 3 -volume,

V_{x} \equiv Δ \bar{x} Δ \bar{y}

Δ \bar{z}

, containing him. Also select a small " 3 -volume in momentum space,

^{'} V_{\bar{p}} \equiv Δ p^{\bar{x}} Δ p^{\bar{y}} Δ p^{\bar{z}}

centered on John's momentum, which is

P^{\bar{x}} = P^{\bar{y}} =

P^{\bar{z}} = 0

.Focus attention on all particles whose world lines pass through

V_{\bar{x}}

and which have momenta

p^{\bar{j}}

in the range

V_{\bar{p}}

surrounding

P^{\bar{j}} = 0

Examine this bundle in another local Lorentz frame ("unbarred frame",

S

) at

P_{0}

, which moves with speed

β

relative to the rest frame. Orient axes so the relative motion of the frames is in the

x

and

\bar{x}

directions. Then the space volume

V_{x}

occupied in the new frame has

Δ y = Δ \bar{y}, Δ z = Δ \bar{z}

(no effect of motion on transverse directions), and

Δ x = {(1 - β^{2})}^{1 / 2} Δ \bar{x}

(Lorentz contraction in longitudinal direction). Hence

V_{x} = {(1 - β^{2})}^{1 / 2} V_{\bar{x}}

("transformation law for space volumes") or, equivalently [since

P^{0} = m / {(1 - β^{2})}^{1 / 2}]

P^{0} V_{x} = m V_{\bar{x}} = (\binom{constant, independent}{of Lorentz frame})

A momentum-space diagram, analogous to the spacetime diagram, depicts the momentum spread for particles in the bundle, and shows that

Δ p^{x} =

Δ p^{\bar{x}} / {(1 - β^{2})}^{1 / 2}

. The Lorentz transformation from

\bar{S}

S

leaves transverse components of momenta unaffected; so

Δ p^{y} = Δ p^{\bar{y}}, Δ p^{z} = Δ p^{\bar{z}}

. Hence

V_{p} = V_{\bar{p}} / {(1 - β^{2})}^{1 / 2}

("transformation law for momentum volumes"); or, equivalently

\frac{V_{p}}{P^{0}} = \frac{V_{\bar{p}}}{m} = (\binom{constant, independent}{of Lorentz frame})

Although the spatial 3 -volumes

V_{x}

and

V_{\bar{x}}

differ from one frame to another, and the momentum 3-volumes

V_{p}

and

V_{\bar{p}}

differ, the volume in six-dimensional phase space is Lorentz-invariant:

V \equiv V_{\bar{x}} T_{\bar{p}} = V_{x} V_{p}

It is a frame-independent, geometric object!

B. For Swarm of Identical Particles with Zero Rest Mass

Examine a sequence of systems, each with particles of smaller rest mass and of higher velocity relative to a laboratory. For every bundle of particles in each system,

P^{0} V_{x}, V_{p} / P^{0}

, and

V_{x} V_{p}

are Lorentzinvariant. Hence, in the limit as

m ⟶ 0

, as

β ⟶ 1

, and as

P^{0} = m / {(1 - β^{2})}^{1 / 2} ⟶

finite value (particles of zero rest mass moving with speed of light),

P^{0} V_{x}

and

V_{p} / P^{0}

and

V_{x} V_{p}

are still Lorentz-invariant, geometric quantities.

Box 22.6 CONSERVATION OF VOLUME IN PHASE SPACE

Examine a very small bundle of identical particles that move through curved spacetime on neighboring geodesics. Measure the bundle's volume in phase space,

V (V = V_{x} V_{p}

in any local Lorentz frame), as a function of affine parameter

λ

along the central geodesic of the bundle. The following calculation shows that

d V / d λ = 0 (\binom{"Liouville theorem in}{curved spacetime"}) .

Proof for particles of finite rest mass: Examine particle motion during time interval

δ τ

, using local Lorentz rest frame of central particle. All velocities are small in this frame, so

p^{\bar{j}} = m d x^{\bar{j}} / d \bar{t}

Hence (see pictures) the spreads in momentum and position conserve

Δ \bar{x} Δ p^{\bar{x}}, Δ \bar{y} Δ p^{\bar{y}}

, and

Δ \bar{z} Δ p^{\bar{z}}

; i.e.,

\frac{d V}{d τ} = \frac{δ (Δ \bar{x} Δ \bar{y} Δ \bar{z} Δ p^{\bar{x}} Δ p^{\bar{y}} Δ p^{\bar{z}})}{δ \bar{t}} = 0

But

τ = a λ + b

for some arbitrary constants

a

and

b

; so

d V / d λ = 0

Proof for particles of zero rest mass. Examine particle motion in local Lorentz frame where central particle has

P = P^{0} (e_{0} + e_{x})

. In this frame, all particles have

p^{y} ≪ p^{0}, p^{z} ≪ p^{0}, p^{x} = p^{0} +

O ({[p^{y}]}^{2} / P^{0}) \approx P^{0}

. Since

p^{α} = d x^{α} / d λ

for appropriate normalization of affine parameters (see Box 22.4), one can write

d x^{j} / d t = p^{j} / p^{0}

; i.e.,

\begin{aligned} \frac{d x}{d t} & = 1 + O ({[p^{y} / P^{0}]}^{2} + {[p^{z} / P^{0}]}^{2}) \\ \approx 1 \\ \frac{d y}{d t} & = \frac{p^{y}}{P^{0}}, \frac{d z}{d t} = \frac{p^{z}}{P^{0}} \end{aligned}

\bar{t} = 0

\bar{t} = δ \bar{t}

Each particle moves with speed

d \bar{x} / d \bar{t}

proportional to height in diagram

d \bar{x} / d \bar{t} = p^{\bar{x}} / m

and conserves its momentum,

d p^{\bar{r}} / d \bar{t} = 0

. Hence the region occupied by particles deforms, but maintains its area. Same is true for

(y - p^{y})

and

(z - p^{z})

Each particle ("photon") moves with

d x / d t = 1

and

d p^{x} / d t = 0

in the local Lorentz frame. Area and shape of occupied region are preserved.

Hence (see pictures)

Δ x Δ p^{x}, Δ y Δ p^{y}

, and

Δ z Δ p^{z}

are all conserved; and

\frac{d V}{d t} = \frac{δ (Δ x Δ y Δ z Δ p^{x} Δ p^{y} Δ p^{z})}{δ t} = 0 .

But

t

and the affine parameter

λ

of central particle are related by

t = P^{0} λ

[cf. equation (16.4)]; thus

d V / d λ = 0

Particle ("photon") speeds are proportional to height in diagram

d y / d t = p^{y} / P^{0}

and

d p^{y} / d t = 0

. Hence, occupied region deforms but maintains its area. Same is true of

z - p^{z}

p (λ)

. Examine the density in phase space in this particle's neighborhood at each point along its world line:

X = R [P (λ), p (λ)] .

Calculate

T (λ)

as follows: (1) Pick an initial event

P (0)

on the world line, and a phase-space volume

V

containing the particle. (2) Cover with red paint all the particles contained in

V

P (0)

. (3) Watch the red particles move through spacetime alongside the initial particle. (4) As they move, the phase-space region they occupy changes shape extensively; but its volume

V

remains fixed (Liouville's theorem). Moreover, no particles can enter or leave that phase-space region (once in, always in; once out, always out; boundaries of phase-space region are attached to and move with the particles). (5) Hence, at any

λ

along the initial particle's world line, the particle is in a phase-space region of unchanged volume

V

, unchanged number of particles

N

, and unchanged ratio

r = N / V

\begin{matrix} (22.47) & \frac{d R [P (λ), p (λ)]}{d λ} = 0 . \end{matrix}

Collisionless Boltzmann equation (kinetic equation)

This equation for the conservation of

Z

along a particle's trajectory in phase space is called the "collisionless Boltzmann equation," or the "kinetic equation."

Photons provide an important application of the Boltzmann equation. But when discussing photons one usually does not think in terms of the number density in phase space. Rather, one speaks of the "specific intensity"

I_{ν}

of radiation at a given frequency

ν

, flowing in a given direction,

n

, as measured in a specified local Lorentz frame:

\begin{matrix} (22.48) & I_{ν} \equiv \frac{d (energy)}{d (time) d (area) d (frequency) d (solid angle)} \end{matrix}

Distribution function for photons expressed in terms of specific intensity,

I_{p}

Invariance and conservation of

I_{p} / ν^{3}

(See Figure 22.2). A simple calculation in the local Lorentz frame reveals that

\begin{matrix} (22.49) & ϰ = h^{- 4} (I_{ν} / ν^{3}), \end{matrix}

where

h

is Planck's constant (see Figure 22.2). Thus, if two different observers at the same or different events in spacetime look at the same photon (and neighboring photons) as it passes them, they will see different frequencies

ν

("doppler shift," "cosmological red shift," "gravitational redshift"), and different specific intensities

I_{ν}

; but they will obtain identical values for the ratio

I_{ν} / ν^{3}

. Thus

I_{p} / ν^{3}

, like

R

, is invariant from observer to observer and from event to event along a given photon's world line.

EXERCISES

Exercise 22.15. INVERSE SQUARE LAW FOR FLUX

The specific flux of radiation entering a telescope from a given source is defined by

\begin{matrix} (22.50) & F_{v} = \int I_{v} d Ω \end{matrix}

where integration is over the total solid angle (assumed

≪ 4 π

) subtended by the source on the observer's sky. Use the Boltzmann equation (conservation of

I_{ν} / ν^{3}

) to show that

F_{v} \propto (distance from source)^{- 2}

for observers who are all at rest relative to each other in flat spacetime.

Exercise 22.16. BRIGHTNESS OF THE SUN

Does the surface of the sun look any brighter to an astronaut standing on Mercury than to a student standing on Earth?

Exercise 22.17. BLACK BODY RADIATION

An "optically thick" source of black-body radiation (e.g., the surface of a star, or the hot matter filling the universe shortly after the big bang) emits photons isotropically with a specific intensity, as seen by an observer at rest near the source, given (Planck radiation law) by

\begin{matrix} (22.51) & I_{ν} = \frac{2 h v^{3}}{e^{h ν / k T} - 1} \end{matrix}

Here

T

is the temperature of the source. Show that any observer, in any local Lorentz frame, anywhere in the universe, who examines this radiation as it flows past him, will also see a black-body spectrum. Show, further, that if he calculates a temperature by measuring the specific intensity

I_{ν}

at any one frequency, and if he calculates a temperature from the shape of the spectrum, those temperatures will agree. (Radiation remains black body rather than being "diluted" into "grey-body.") Finally, show that the temperature he measures is redshifted by precisely the same factor as the frequency of any given photon is redshifted,

\begin{matrix} (22.52) & \frac{T_{observed}}{T_{emitted}} = (\frac{v_{observed}}{v_{emitted}}) for a given photon. \end{matrix}

[Note that the redshifts can be "Doppler" in origin, "cosmological" in origin, "gravitational" in origin, or some inseparable mixture. All that matters is the fact that the parallel-transport law for a photon's 4-momentum,

\nabla_{p} p = 0

, guarantees that the redshift

ν_{observed} / v_{emitted}

is independent of frequency emitted.]

3 -momentum volume, with direction of momentum vectors reversed for ease of visualization (telescope as an emitter, not a receiver!)
Figure 22.2.
Number density in phase space for photons, interpreted in terms of the specific intensity

I_{y}

. An astronomer has a telescope with filter that admits only photons arriving from within a small solid angle

Δ Ω

about the

z

-direction, and having energies between

p^{0}

and

p^{0} + Δ p^{0}

. The collecting area,

a

, of his telescope lies in the

x, y

-plane (perpendicular to the incoming photon beam). Let

δ N

be the number of photons that cross the area

a

in a time interval

δ t

. [All energies, areas, times, and lengths are measured in the orthonormal frame ("proper reference frame; §13.6) which the astronomer Fermi-Walker transports with himself along his (possibly accelerated) world line-or, equivalently, in a local Lorentz frame momentarily at rest with respect to the astronomer.] The

δ N

photons, just before the time interval

δ t

begins, lie in the cylinder of area

a

and height

δ z = δ t

shown above. Their spatial 3 -volume is thus

V_{x} = a δ t

. Their momentum 3-volume is

V_{p} = {(p^{0})}^{2} Δ p^{0} Δ Ω

(see drawing). Hence, their number density in phase space is

R = \frac{δ N}{V_{x} {\tilde{V}}_{p}} = \frac{δ N}{a δ t {(p^{0})}^{2} (Δ p^{0}) Δ Ω} = \frac{δ N}{h^{3} G δ t v^{2} Δ v Δ Ω}

where

ν

is the photon frequency measured by the telescope (

p^{0} = h ν

).
The specific intensity of the photons,

I_{p}

(a standard concept in astronomy), is the energy per unit area per unit time per unit frequency per unit solid angle crossing a surface perpendicular to the beam: i.e.,

I_{ν} = \frac{h v δ N}{a δ t Δ v Δ Ω}

Direct comparison reveals

π = h^{- 4} (I_{ν} / ν^{3})

.
Thus, conservation of

r

along a photon's world line implies conservation of

I_{y} / ν^{3}

. This conservation law finds important applications in cosmology (e.g., Box 29.2 and Ex. 29.5) and in the gravitational lens effect (Refsdal 1964); see also exercises 22.15-22.17.

Exercise 22.18. STRESS-ENERGY TENSOR

(a) Show that the stress-energy tensor for a swarm of identical particles at an event

P_{0}

can be written as an integral over the mass hyperboloid of the momentum space at

P_{0}

\begin{matrix} (22.53) & T = \int (T p \otimes p) (d V_{p} / p^{0}) \\ (22.54) & \frac{d V_{p}}{p^{0}} \equiv \frac{d p^{x} d p^{y} d p^{z}}{p^{0}} in a local Lorentz frame. \end{matrix}

(Notice from Box 22.5 that

d V_{p} / p^{0}

is a Lorentz-invariant volume element for any segment of the mass hyperboloid.)
(b) Verify that the Boltzmann equation,

d π / d λ = 0

, implies

\nabla \cdot T = 0

for any swarm of identical particles. [Hint: Calculate

\nabla \cdot T

in a local Lorentz frame, using the above expression for

T

, and using the geodesic equation in the form

D p^{μ} / d λ = 0

Exercise 22.19. KINETIC THEORY FOR NONIDENTICAL PARTICLES

For a swarm of particles with a wide distribution of rest masses, define

\begin{matrix} (22.55) & R = \frac{Δ N}{V_{x} V_{p} Δ m}, \end{matrix}

where

V_{x}

and

V_{p}

are spatial and momentum 3-volumes, and

Δ N

is the number of particles in the region

V_{x} V_{p}

with rest masses between

m - Δ m / 2

and

m + Δ m / 2

. Show the following.
(a)

T_{x} V_{p} Δ m

is independent of Lorentz frame and independent of location on the world tube of a bundle of particles.
(b)

R

can be regarded as a function of location

P

in spacetime and 4-momentum

p

inside the future light cone of the tangent space at

P

\begin{matrix} (22.56) & R = H (P, p) . \end{matrix}

(c)

π

satisfies the collisionless Boltzmann equation (kinetic equation)

\begin{matrix} (22.57) & \frac{d H [P (λ), p (λ)]}{d λ} = 0 along geodesic trajectory of any particle. \end{matrix}

(d)

R

can be rewritten in a local Lorentz frame as

\begin{matrix} (22.58) & ℜ = \frac{Δ N}{[(p^{0} / m) Δ x Δ y Δ z] [Δ p^{0} Δ p^{x} Δ p^{y} Δ p^{z}]} \end{matrix}

(e) The stress-energy tensor at an event

P

can be written as an integral over the interior of the future light cone of momentum space

\begin{matrix} (22.59) & T^{μ ν} = \int (ℜ p^{μ} p^{ν}) m^{- 1} d p^{0} d p^{1} d p^{2} d p^{3} \end{matrix}

in a local Lorentz frame (Track-1 notation for integral; see Box 5.3);

\begin{aligned} T & = \int (\overset{⏜}{τ} p \otimes p) m^{- 1 *} 1 in frame-independent notation \\ (') & = \int (\overset{⏜}{} p \otimes p) m^{- 1} d p^{0} \land d p^{1} \land d p^{2} \land d p^{3} \end{aligned}

in a local Lorentz frame (Track-2 notation; see Box 5.4).

RELATIVISTIC STARS

Wherein the reader, armed with the magic potions and powers of Geometrodynamics, conquers the stars.

CHAPTER 3 SPHERICAL STARS

§23.1. PROLOG

Beautiful though gravitation theory may be, it is a sterile subject until it touches the real physical world. Only the hard reality of experiments and of astronomical observations can bring gravitation theory to life. And only by building theoretical models of stars (Part V), of the universe (Part VI), of stellar collapse and black holes (Part VII), of gravitational waves and their sources (Part VIII), and of gravitational experiments (Part IX), can one understand clearly the contacts between gravitation theory and reality.
The model-building in this book will follow the tradition of theoretical physics. Each Part (stars, universe, collapse, . . .) will begin with the most oversimplified model conceivable, and will subsequently add only those additional touches of realism necessary to make contact with the least complex of actual physical systems. The result will be a tested intellectual framework, ready to support and organize the additional complexities demanded by greater realism. Greater realism will not be attempted in this book. But the reader seeking it could start in no better place than the two-volume treatise on Relativistic Astrophysics by Zel'dovich and Novikov (1971, 1974).
Begin, now, with models for relativistic stars. As a major simplification, insist (initially) that all stars studied be static. Thereby exclude not only exploding and pulsating stars, but even quiescent ones with stationary rotational motions. From the static assumption, plus a demand that the star be made of "perfect fluid" (no shear stresses allowed!), plus Einstein's field equations, it probably follows that the star is spherically symmetric. However, nobody has yet given a proof. [For proofs under more restricted assumptions, see Avez (1964) and Kunzle (1971).] In the absence of a proof, assume the result: insist that all stars studied be spherical as well as static.

Preview of the rest of this book

Static stars must be spherical

§23.2. COORDINATES AND METRIC FOR A STATIC, SPHERICAL SYSTEM

Metric for any static, spherical system:
(1) generalized from flat spacetime
(2) specialized to
"Schwarzschild form"

To deduce the gravitational field for a static spherical star-or for any other static, spherical system-begin with the metric of special relativity (no gravity) in the spherically symmetric form

\begin{matrix} (23.1) & d s^{2} = - d t^{2} + d r^{2} + r^{2} d Ω^{2} \end{matrix}

where

\begin{matrix} (23.2) & d Ω^{2} = d θ^{2} + \sin^{2} θ d ϕ^{2} \end{matrix}

Try to modify this metric to allow for curvature due to the gravitational influence of the star, while preserving spherical symmetry. The simplest and most obvious guess is to allow those metric components that are already non-zero in equation (23.1) to assume different values:

\begin{matrix} (23.3) & d s^{2} = - e^{2 ϕ} d t^{2} + e^{2 Λ} d r^{2} + R^{2} d Ω^{2} \end{matrix}

where

Φ, Λ

, and

R

are functions of

r

only. (The static assumption demands

\partial g_{μ ν} / \partial t = 0

.) To verify that this guess is good, use it in constructing stellar models, and check that the resulting models have the same generality (same set of quantities freely specifiable) as in Newtonian theory and as expected from general physical considerations. An apparently more general metric

\begin{matrix} (23.4) & d s^{2} = - a^{2} d t^{2} - 2 a b d r d t + c^{2} d r^{2} + R^{2} d Ω^{2} \end{matrix}

actually is not more general in any physical sense. One can perform a coordinate transformation to a new time coordinate

t^{'}

defined by

\begin{matrix} (23.5) & e^{⧸} d t^{'} = a d t + b d r \end{matrix}

By inserting this in equation (23.4), and by defining

e^{2 A} \equiv b^{2} + c^{2}

, one obtains the postulated line element (23.3), apart from a prime on the

t .^{*}

The necessity to allow for arbitrary coordinates in general relativity may appear burdensome when one is formulating the theory; but it gives an added flexibility, something one should always try to turn to one's advantage when formulating and solving problems. The

g_{r t} = 0

simplification (called a coordinate condition) in equation (23.3) results from an advantageous choice of the

t

coordinate. The

r

coordinate, however, is also at one's disposal (as long as one chooses it in a way that respects spherical symmetry; thus not

r^{'} = r + \cos θ

). One can turn this freedom to advantage by introducing a new coordinate

r^{'} (r)

defined by

\begin{matrix} (23.6) & r^{'} = R (r) \end{matrix}

With this choice of the radial coordinate, and with the primes dropped, equation (23.3) reduces to

\begin{matrix} (23.7) & d s^{2} = - e^{2 Φ} d t^{2} + e^{2 Λ} d r^{2} + r^{2} d Ω^{2} \end{matrix}

a line element with just two unknown functions,

Φ (r)

and

Λ (r)

. This coordinate system and metric have been used in most theoretical models for relativistic stars since the pioneering work of Schwarzschild (1916b), Tolman (1939), and Oppenheimer and Volkoff (1939). These particular coordinates are sometimes called "curvature coordinates" and sometimes "Schwarzschild coordinates." The central idea of these coordinates, in a nutshell, is (Schwarzschild

r

-coordinate)

=

(proper circumference)

/ 2 π

For a more rigorous proof that in any static spherical system Schwarzschild coordinates can be introduced, bringing the metric into the simple form (23.7), see Box 23.3 at the end of this chapter.

Exercise 23.1. ISOTROPIC COORDINATES AND NEWTONIAN LIMIT

An alternative set of coordinates sometimes used for static, spherical systems is the "isotropic coordinate system"

(t, \bar{r}, θ, ϕ)

. The metric in isotropic coordinates has the form

\begin{matrix} (23.8) & d s^{2} = - e^{2 ϕ} d t^{2} + e^{2 μ} [d {\bar{r}}^{2} + {\bar{r}}^{2} d Ω^{2}] \end{matrix}

with

Φ

and

μ

being functions of

\bar{r}

.
(a) Exhibit the coordinate transformation connecting the Schwarzschild coordinates (23.7) to the isotropic coordinates (23.8).
(b) From equation (16.2a) [or equivalently (18.15c)], show that, in the Newtonian limit, the metric coefficient

Φ

of the isotropic line element becomes the Newtonian potential; and

μ

becomes equal to

- Φ

. By combining with part (a), discover that

Λ = r d Φ / d r

in the Newtonian limit.

EXERCISE

§23.3. PHYSICAL INTERPRETATION OF SCHWARZSCHILD COORDINATES

In general relativity, because the use of arbitrary coordinates is permitted, the physical significance of statements about tensor or vector components and other quantities is not always obvious. There are, however, some situations where the interpretation is almost as straightforward as in special relativity. The most obvious example is the center point of a local inertial coordinate system, where the principle of equivalence allows one to treat all local quantities (quantities not involving spacetime curvature) exactly as in special relativity. Schwarzschild coordinates for a spherical system turn out to be a second example.

One's first reaction when meeting a new metric should be to examine it, not in order to learn about the gravitational field, for which the curvature tensor is more

The form of any metric can reveal the nature of the coordinates being used

Geometric significance of the Schwarzschild coordinates:
(1)

θ, ϕ

are angles on sphere
(2)

r

measures surface area of sphere
(3)

t

has 3 special geometric properties
(4) description of a "machine" to measure

t

directly informative, but to learn about the coordinates. (Are they, for instance, locally inertial at some point?)
The names given to the coordinates have no intrinsic significance. A coordinate transformation

t^{'} = θ, r^{'} = ϕ, θ^{'} = r, ϕ^{'} = t

is perfectly permissible, and has no influence on the physics or the mathematics of a relativistic problem. The only thing it affects is easy communication between the investigator who adopts it and his colleagues. Thus the names

tr θ ϕ

for the Schwarzschild coordinates (23.7) provide a mnemonic device pointing out the geometric content of the coordinates.* In particular, the names

θ, ϕ

are justified by the fact that on each two-dimensional surface of constant

r

and

t

, the distance between two nearby events is given by

d s^{2} = r^{2} d Ω^{2}

, as befits standard

θ, ϕ

coordinates on a sphere of radius

r

. The area of this two-dimensional sphere is clearly

\begin{matrix} (23.9) & A = \int (r d θ) (r \sin θ d ϕ) = 4 π r^{2} \end{matrix}

hence, the metric (23.7) tells how to measure the

r

coordinate that it employs. One can merely measure (in proper length units) the area

A

of the sphere, composed of all points rotationally equivalent to the point

P

for which the value

r (P)

is desired; and one can then calculate

\begin{matrix} (') & r (P) = {(\begin{array}{l} proper area of sphere \\ through point P \end{array} / 4 π)}^{1 / 2} . \end{matrix}

The Schwarzschild coordinates have been picked for convenience, and not for the ease with which one could build a coordinate-measuring machine. This makes it more difficult to design a machine to measure

t

than machines to measure

r, θ, ϕ

.
The geometric properties of

t

on which a measuring device can be based are: (1) the time-independent distances (

\partial g_{α β} / \partial t = 0

) between world lines of constant

r, θ, ϕ; (2)

the orthogonality

(g_{t r} = g_{t θ} = g_{t ϕ} = 0)

of these world lines to the

t =

constant hypersurfaces; and (3) a labeling of these hypersurfaces by Minkowski (special relativistic) coordinate time at spatial infinity, where spacetime becomes flat. This labeling produces a constraint

\begin{matrix} (23.10) & Φ (\infty) = 0 \end{matrix}

in the metric (23.7). [Mathematically, this constraint is imposed by a simple rescaling transformation

t^{'} = e^{Φ (x)} t

, and by then dropping the prime.]
One "machine" design which constructs (mentally) such a

t

coordinate, and in the process measures it, is the following. Observers using radar sets arrange to move along the coordinate lines

r, θ, ϕ =

const. They do this by adjusting their velocities until each finds that the radar echos from his neighbors, or from "benchmark" reference points in the asymptotically flat space, require the same round-trip time at each repetition. Equivalently, each returning echo must show zero doppler shift;

\begin{aligned} *For an example of misleading names, consider those in the equation \\ d s^{2} = - e^{2 ϕ (θ)} d ϕ^{' 2} + e^{2 A (θ^{'})} d θ^{' 2} + θ^{' 2} (d t^{' 2} + \sin^{2} t^{'} d r^{' 2}), \end{aligned}

which is equivalent to equation (23.7), but employs the coordinates

t^{'} = θ, r^{'} = ϕ, θ^{'} = r, ϕ^{'} = t

.
it must return with the same frequency at which it was sent out. Next a master clock is set up near spatial infinity (far from the star). It is constructed to measure proper time-which, for it, is Minkowski time "at infinity"-and to emit a standard oneHertz signal. Each observer adjusts the rate of his "coordinate clock" to beat in time with the signals he receives from the master clock. To set the zero of his "coordinate clock," now that its rate is correct, he synchronizes with the master clock, taking account of the coordinate time

Δ t

required for radar signals to travel from the master to him. [To compute the transit time, he assumes that for radar signals (

t_{reflection} -

t_{emission}) = (t_{return} - t_{reflection}) = Δ t

, so that the echo is obtained by time-inversion about the reflection event. This time-reversal invariance distinguishes the time

t

in the metric (23.7) from the more general

t

coordinates allowed by equation (23.4).] Each observer moving along a coordinate line (

r, θ, ϕ =

const.) now has a clock that measures coordinate time

t

in his neighborhood.

The above discussion identifies the Schwarzschild coordinates of equation (23.7) by their intrinsic geometric properties. Not only are

r

and

t

radial and time variables, respectively (in that

\partial / \partial r

and

\partial / \partial t

are spacelike and timelike, respectively, and are orthogonal also to the spheres defined by rotational symmetry), but they have particular properties

[4 π r^{2} =

surface area;

\partial g_{μ ν} / \partial t = 0; \partial / \partial r \cdot \partial / \partial t = g_{r t} = 0

;

\partial / \partial t \cdot \partial / \partial t = g_{t t} = - 1

r = \infty

] that distinguish them from other possible coordinate choices

[r^{'} = f (r), t^{'} = t + F (r)]

. No claim is made that these are the only coordinates that might reasonably be called

r

and

t

; for an alternative choice ("isotropic coordinates"), see exercise 23.1. However, they provide a choice that is reasonable, vnambiguous, useful, and often used.

§23.4. DESCRIPTION OF THE MATTER INSIDE A STAR

To high precision, the matter inside any star is a perfect fluid. (Shear stresses are negligible, and energy transport is negligible on a "hydrodynamic time scale.") Thus, it is reasonable in model building to describe the matter by perfect-fluid parameters:

\begin{aligned} ρ & = ρ (r) = density of mass-energy in rest-frame of fluid; \\ p & = p (r) = isotropic pressure in rest-frame of fluid; \\ (23.11) & n & = n (r) = number density of baryons in rest-frame of fluid; \\ u^{μ} & = u^{μ} (r) = 4 -velocity of fluid; \\ (23.12) & T^{μ ν} & = (ρ + p) u^{μ} u^{ν} + p g^{μ ν} = stress-energy tensor of fluid. \end{aligned}

(For Track-1 discussion, see Box 5.1; for greater Track-2 detail, see

§ § 22.2

and 22.3.) In order that the star be static, each element of fluid must remain always at rest in the static coordinate system; i.e., each element must move along a world line of constant

r, θ, ϕ

; i.e., each element must have 4 -velocity components

\begin{matrix} (23.13a) & u^{r} = d r / d τ = 0, u^{θ} = d θ / d τ = 0, u^{ϕ} = d ϕ / d τ = 0 \end{matrix}

Material inside star to be
idealized as perfect fluid

Parameters describing perfect fluid:
(1)

ρ, p, n

(2)

u

(1)

Other coordinates are possible, but Schwarzschild are particularly simple

The normalization of 4 -velocity,

- 1 = u \cdot u = g_{μ ν} u^{μ} u^{ν} = g_{t t} u^{t} u^{t} = - e^{2 Φ} u^{t} u^{t}

then determines

u^{t}

\begin{matrix} (23.13b) & u^{t} = d t / d τ = e^{- Φ}, u = e^{- Φ} \partial / \partial t \end{matrix}

and this, together with the general form (23.12) of the stress-energy tensor and the form (23.7) of the metric, determines

T^{μ ν}

\begin{matrix} (23.14) & T^{00} = ρ e^{- 2 ϕ}, T^{r r} = p e^{- 2 Λ}, T^{θ θ} = p r^{- 2}, T^{ϕ ϕ} = p r^{- 2} \sin^{- 2} θ, \\ T^{α β} = 0 if α \neq β . \end{matrix}

Although these components of the stress-energy tensor in Schwarzschild coordinates are useful for calculations, the normalization factors

e^{- 2 Φ}, e^{- 2 Λ}, r^{- 2}, r^{- 2} \sin^{- 2} θ

make them inconvenient for physical interpretations. More convenient are components on orthonormal tetrads carried by the fluid elements ("proper reference frames"; see §13.6):

Proper reference frame of fluid

Components of

u

and

T

in proper reference frame

Equation of state:
(1) in general
(2) idealized to "one-parameter form"

p = p (n), ρ = ρ (n)

\begin{matrix} (23.15d) & \begin{array}{r} e_{\hat{t}} \equiv \frac{d}{d τ} = \frac{1}{e^{ϕ}} \frac{\partial}{\partial t}, e_{\hat{r}} = \frac{1}{e^{A}} \frac{\partial}{\partial r}, e_{\hat{θ}} = \frac{1}{r} \frac{\partial}{\partial θ}, e_{\hat{ϕ}} = \frac{1}{r \sin θ} \frac{\partial}{\partial ϕ}; \\ ω^{\hat{t}} = e^{ϕ} d t, w^{\hat{r}} = e^{Λ} d r, w^{\hat{θ}} = r d θ, ω^{\hat{ϕ}} = r \sin θ d ϕ \\ u = e_{\hat{t}}; u^{\hat{t}} = 1, u^{\hat{r}} = u^{\hat{θ}} = u^{\hat{ϕ}} = 0; \\ T_{\hat{i t} \hat{t}} \equiv T_{\hat{0} \hat{0}} = ρ, T_{\hat{r} \hat{r}} = T_{\hat{θ} \hat{θ}} = T_{\hat{ϕ} \hat{ϕ}} = p, T_{\hat{α} \hat{β}} = 0 if α \neq β . \end{array} \end{matrix}

See exercise 23.2 below.
The structure of a star-i.e., the set of functions

Φ (r), Λ (r), ρ (r), p (r), n (r)

-is determined in part by the Einstein field equations,

G^{μ ν} = 8 π T^{μ ν}

, and in part by the law of local conservation of energy-momentum in the fluid,

T^{μ ν}_{; ν} = 0

. However, these are not sufficient to fix the structure uniquely. Also necessary is the functional dependence of pressure

p

and density

ρ

on number density of baryons

n

\begin{matrix} (23.16) & p = p (n), ρ = ρ (n) \end{matrix}

Normally one cannot deduce

p

and

ρ

from a knowledge solely of

n

. One must know, in addition, the temperature

T

or the entropy per baryon

s

; then the laws of thermodynamics plus equations of state will determine all remaining thermodynamic variables:

p = p (n, s), ρ = ρ (n, s), \dots

(See

§ 22.2

and Box 22.1 for full Track-2 discussions.) To pass from the given thermodynamic knowledge,

p (n, s)

and

ρ (n, s)

, to the desired knowledge,

p (n)

and

ρ (n)

, one needs information about the star's thermal properties, and especially about the way in which energy generation plus heat flow have conspired to distribute the entropy,

s = s (n)

p (n) = p [n, s (n)], ρ (n) = ρ [n, s (n)] .

There exist three important applications of the theory of relativistic stars: neutron stars, white dwarfs, and supermassive stars (stars with

M \geq 10^{3} M_{⊙}

, which may exist according to theory, but the existence of which has never yet been confirmed by observation). In all three cases, happily, the passage from

p = p (n, s), ρ (n, s)

, to

p = p (n), ρ = ρ (n)

, is trivial.

Consider first a neutron star. Though hot by ordinary standards, a neutron star is so cold by any nuclear-matter scale of temperatures that essentially all its thermal degrees of freedom are frozen out ("degenerate gas"; "quantum fluid"). It is not important that a detailed treatment of the substance of a neutron star is beyond the capability of present theory (allowance for the interaction between baryon and baryon; production at sufficiently high pressures of hyperons and mesons). The simple fact is that one is dealing with matter at densities comparable to the density of matter in an atomic nucleus

(2 \times 10^{14} g / {cm}^{3})

and higher. Everything one knows about nuclear matter [see, for example, Bohr and Mottelson (1969)] tells one that it is degenerate, and that one can estimate in order of magnitude its degeneracy temperature by treating it as though it were an ideal Fermi neutron gas. (In a normal atomic nucleus, a little more than 50 per cent of all baryons are neutrons, the rest are protons; in a neutron star, as many as 99 per cent are neutrons.) When approximating the neutron-star matter as an ideal Fermi neutron gas, one considers the neutrons to occupy free-particle quantum states, with two particles of opposite spin in each occupied state, and a sharp drop from 100 per cent occupancy of quantum states to empty states when the particle energy rises to the level of the "Fermi energy" [for more on such an ideal Fermi gas, see Kittel, Section 19 (1958); or at an introductory level, see Sears, Section 16-5 (1953)]. In matter at nuclear density, the Fermi energy is of the order

E_{Fermi} \sim 30 MeV or 3 \times 10^{11} K

and at higher density the temperature required to unfreeze the degeneracy is even greater. In other words, for matter at and above nuclear densities, already at zero temperature the kinetic energy of the particles (governed by the Pauli exclusion principle and by their Fermi energy) is a primary source of pressure. Nuclear forces make a large correction to this pressure, but for

T ≦ 30 MeV = 3 \times 10^{11} K

, energies of thermal agitation do not.

A star, in collapsing from a normal state to a neutron-star state (see Chapter 24), emits a huge flux of neutrinos at temperatures

\geq 10^{10} K

, and thereby cools to

T ≪ 3 \times 10^{11} K

within a few seconds after formation. Consequently, in all neutron stars older than a few seconds one can neglect thermal contributions to the pressure and density; i.e., one can set

p (n, s) = p (n, s = 0) = p (n), ρ (n, s) = ρ (n, s = 0) = ρ (n)

A white dwarf is similar, except that here electrons rather than neutrons are the source of Fermi gas pressure and degeneracy. Typical white-dwarf temperatures satisfy

k T ≪ E_{Fermi electrons};

Justification for idealized equation of state:
(1) in neutron stars
(2) in white dwarfs
(3) in supermassive stars
the Fermi kinetic energy (Pauli exclusion principle), and not random

k T

energy, is primarily responsible for the pressure and energy density; and one can set

p (n, s) = p (n, s = 0) = p (n), ρ (n, s) = ρ (n, s = 0) = ρ (n) .

In a supermassive star (see Chapter 24), the situation is quite different. There temperature and entropy are almost the whole story, so far as pressure and energy density are concerned. However, convection keeps the star stirred up and produces a uniform entropy distribution

s = const. independent of radius;

so one can write

\begin{array}{r} p (n, s) = p_{s} (n), ρ (n, s) = ρ_{s} (n) . \\ [\begin{array}{l} functions depending on \\ uniform entropy per baryon, \\ s, in the star \end{array}] ↑ \end{array}

In all three cases-neutron stars, white dwarfs, supermassive stars-one regards the relations

p (n)

and

ρ (n)

as "equations of state"; and having specified them, one can calculate the star's structure without further reference to its thermal properties.

EXERCISE

Exercise 23.2. PROPER REFERENCE FRAMES OF FLUID ELEMENTS

(a) Verify that equations

(23.15 a, b)

define an orthonormal tetrad and its dual basis of 1 -forms, at each event in spacetime.
(b) Verify that the components of the fluid 4 -velocity relative to these tetrads are given by equations ( 23.15 c ). Why do these components guarantee that the tetrads form "proper reference frames" for the fluid elements?
(c) Verify equations

(23.15 d)

for the components of the stress-energy tensor.

§23.5. EQUATIONS OF STRUCTURE

Five equations needed to determine 5 stellar-structure functions:

Φ, Λ, p, ρ, n

The structure of a relativistic star is determined by five functions of radius

r

: the metric functions

Φ (r), Λ (r)

, the pressure

p (r)

, the density of mass-energy

ρ (r)

, and the number density of baryons,

n (r)

. Hence, to determine the structure uniquely, one needs five equations of structure, plus boundary conditions. Two equations of structure, the equations of state

p (n)

and

ρ (n)

, are already in hand. The remaining three must be the essential content of the Einstein field equations and of the law of local energy-momentum conservation,

T^{μ ν}_{; ν} = 0

One knows that the law of local energy-momentum conservation for the fluid follows as an identity from the Einstein field equations. Without loss of information,
one can therefore impose all ten field equations and ignore local energy-momentum conservation. But that is an inefficient way to proceed. Almost always the equations

T^{μ ν}_{; ν} = 0

can be reduced to usable form more easily than can the field equations. Hence, the most efficient procedure is to: (1) evaluate the four equations

T^{μ ν}_{; ν} = 0

; (2) evaluate enough field equations (six) to obtain a complete set

(6 + 4 = 10)

; and (3) evaluate the remaining four field equations as checks of the results of (1) and (2).

The Track-2 reader has learned (

$ 22.3

) that the equations

T^{μ ν}_{; ν} = 0

for a perfect fluid take on an especially simple form when projected (1) on the 4 -velocity

u

of the fluid itself, and (2) orthogonal to

u

. Projection along

u (u_{μ} T^{μ ν}_{; ν} = 0)

gives the local law of energy conservation (22.11a),

\frac{d ρ}{d τ} = - (ρ + p) \nabla \cdot u = \frac{ρ + p}{n} \frac{d n}{d τ}

where

u = d / d τ

; i.e.,

τ

is proper time along the world line of any chosen element of the fluid. For a static star, or for any other static system, both sides of this equation must vanish identically (no fluid element ever sees any change in its own density).
Projection of

T^{μ ν}_{; ν} = 0

orthogonal to

u

gives the reasonable equation

(\binom{inertial mass}{per unit volume}) \times (4 -acceleration) = - (\binom{pressure gradient, projected}{perpendicular to u})

i.e.,

(ρ + p) \nabla_{u} u = - [\nabla p + (\nabla_{u} p) u] .

[see equation (22.13)]. When applied to a static star, this equation tells how much pressure gradient is needed to prevent a fluid element from falling. Only the radial component of this equation has content, since the pressure depends only on

r

. The radial component in the Schwarzschild coordinate system says [see the line element (23.7) and the 4 -velocity components (23.13)],

\begin{aligned} (ρ + p) u_{r; p} u^{ν} & = - (ρ + p) Γ_{r ν}^{α} u_{α} u^{ν} = - (ρ + p) Γ_{r 0}^{0} u_{0} u^{0} \\ (23.17) & = (ρ + p) Φ_{, r} = - p_{, r} \end{aligned}

(Track-1 readers can derive this from scratch at the end of the section, exercise 23.3.) In the Newtonian limit,

Φ

becomes the Newtonian potential (since

g_{00} =

- e^{2 Φ} \approx - 1 - 2 Φ)

, and the pressure becomes much smaller than the mass-energy density; consequently equation (23.17) becomes

\begin{matrix} (23.17~N) & ρ Φ_{, r} = - p_{, r} \end{matrix}

This is the Newtonian version of the equation describing the balance between gravitational force and pressure gradient.

The pressure gradient that prevents a fluid element from falling appears in Einstein's theory as the source of an acceleration. This acceleration, by keeping the fluid element at a fixed

r

value, causes it to depart from geodesic motion (from "fiducial world line"; from motion of free fall into the center of the star). Newtonian

The most efficient procedure

Equation of hydrostatic equilibrium derived
for solving Einstein equations

Comparison of Newton and Einstein views of hydrostatic equilibrium

Equation for $Λ$ derived

"Mass-energy inside radius

r

m (r)

defined

r,^{''} m (r)

, defined
theory, on the other hand, views as the fiducial world line the one that stays at a fixed

r

value. It regards the "gravitational force" as trying (without success, because balanced by the pressure gradient) to pull a particle from a fixed-r world line onto a geodesic world line. In the two theories the magnitudes of the acceleration, whether "actually taking place" (Einstein theory) or "trying to take place" (Newtonian theory), are the same to lowest order (but opposite in direction); so it is no surprise that

(23.17)

and

(23.17 N)

differ only in detail.

Turn next to the Einstein field equation. Here, as is often the case, the components of the field equation in the fluid's orthonormal frame [equations (23.15a,b)] are simpler than the components in the coordinate basis. One already knows the stressenergy tensor

T_{\hat{α} \hat{β}}

in the orthonormal frame [equation (23.15d)]; and Track-2 readers have already calculated the Einstein tensor

G_{\hat{α} \hat{β}}

(exercise 14.13 ; Track-1 readers will face the task at the end of this section, exercise 23.4). All that remains is to equate

G_{\hat{α} \hat{β}}

8 π T_{\hat{α} \hat{β}}

. Examine first the

\hat{0} \hat{0}

component of the field equations:

\begin{aligned} G_{\hat{0} \hat{0}} & = r^{- 2} - r^{- 2} e^{- 2 Λ} - r^{- 1} (d / d r) (e^{- 2 Λ}) \\ = r^{- 2} (d / d r) [r (1 - e^{- 2 Λ})] = 8 π T_{\hat{0} \hat{0}} = 8 π ρ \end{aligned}

This equation becomes easy to solve as soon as one notices that it is a differential equation linear in the quantity

e^{- 2 A}

; a bit of tidying up then focuses attention on the quantity

r (1 - e^{- 2 Λ})

. Give this quantity the name

2 m (r)

(so far only a name!); thus,

\begin{matrix} (23.18) & 2 m \equiv r (1 - e^{- 2 Λ}); e^{2 Λ} = (1 - 2 m / r)^{- 1} \end{matrix}

In this notation the

\hat{0} \hat{0}

component of the Einstein tensor becomes

G_{\hat{0} \hat{0}} = \frac{2}{r^{2}} \frac{d m (r)}{d r} = 8 π ρ

Integrate and find

\begin{matrix} (23.19) & m (r) = \int_{0}^{r} 4 π r^{2} ρ d r + m (0) \end{matrix}

For the constant of integration

m (0)

, a zero value means a space geometry smooth at the origin (physically acceptable); a non-zero value means a geometry with a singularity at the origin (physically unacceptable: no local Lorentz frame at

r = 0

\begin{aligned} d s^{2} & = [1 - 2 m (0) / r]^{- 1} d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \\ \approx - [r / 2 m (0)] d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) at r \approx 0 if m (0) \neq 0 \\ (23.20) & d s^{2} & = {[1 - (8 π / 3) ρ_{c} r^{2}]}^{- 1} d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \\ \approx d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) at r \approx 0 if m (0) = 0 \end{aligned}

The quantity

m (r)

, defined by equation (23.18) and calculated from equation (23.19) with

m (0) = 0

, is a relativistic analog of the "mass-energy inside radius

r

." Box 23.1 spells out the analogy in detail.

Box 23.1 MASS-ENERGY INSIDE RADIUS $r$

The total mass-energy

M

of an isolated star is well-defined (Chapter 19). But not well-defined, in general, is the distribution of that mass-energy from point to point inside the star and in its gravitational field (no unique "gravitational stress-energy tensor"). This was the crucial message of §20.4 (Track 2).

The message is true in general. But for the case of a spherical star-and only for that case-the message loses its bite. Spherical symmetry allows one to select a distribution of the total mass-energy that is physically reasonable. In Schwarzschild coordinates, it is defined by

\begin{matrix} (1) & "total mass-energy inside radius r " \equiv m (r) = \int_{0}^{r} 4 π r^{2} ρ d r . \end{matrix}

The fully convincing argument for this definition is found only by considering a generalization of it to time-dependent spherically symmetric stars (pulsating, collapsing, or exploding stars; see Chapters 26 and 32 , and especially exercise 32.7). For them one finds that the mass-energy

m

associated with a given ball of matter (fixed baryon number) can change in time only to the extent that locally measurable energy fluxes can be detected at the boundary of the ball. [Such energy fluxes could be the power expended by pressure forces against the moving boundary surface, or heat fluxes, or radiation (photon or neutrino) fluxes. But since spherically symmetric gravitational waves do not exist (Chapters 35 and 36), neither physical intuition nor Einstein's equations require that problems of localizing gravitational-wave energy be faced.] Thus the energy

m

is localized, not by a mathematical convention, but by the circumstance that transfer of energy (with this definition of

m

) is detectable by local measurements. [For the mathematical details of

m (r, t)

in the time-dependent case, see Misner and Sharp (1964), Misner (1965), and exercise 32.7.]

In addition to the critical "local energy flux" property of

m (r)

described above, there are three further properties that verify its identification as mass-energy. They are: (1) Everywhere outside the star

\begin{matrix} (2) & m (r) = M \equiv (\binom{total mass-energy of star as measured from}{Kepler's third law for distant planets}) \end{matrix}

see

§ 23.6

for proof. (2) For a Newtonian star, where "mass inside radius

r

" has a unique meaning,

m (r)

is that mass. (3) For a relativistic star,

m (r)

splits nicely into "rest mass-energy"

m_{0} (r)

plus "internal energy"

U (r)

plus "gravitational potential energy"

Ω (r)

To recognize and appreciate the split

\begin{matrix} (3) & m (r) = m_{0} (r) + U (r) + Ω (r), \end{matrix}

proceed as follows. First split the total density of mass-energy,

ρ

, into a part

μ_{0} n

due to rest mass - where

μ_{0}

is the average rest mass of the baryonic species pres-

Box 23.1 (continued)

ent-and a part

ρ - μ_{0} n

due to internal thermal energy, compressional energy, etc. Next notice that the proper volume of a shell of thickness

d r

\begin{matrix} (4) & d V = 4 π r^{2} (e^{A} d r) = 4 π r^{2} (1 - 2 m / r)^{- 1 / 2} d r \end{matrix}

not

4 π r^{2} d r

. Consequently, the total rest mass inside radius

r

\begin{matrix} (5) & m_{0} = \int_{0}^{r} μ_{0} n d V = \int_{0}^{r} 4 π r^{2} (1 - 2 m / r)^{- 1 / 2} μ_{0} n d r \end{matrix}

and the total internal energy is

\begin{matrix} (6) & U = \int_{0}^{r} (ρ - μ_{0} n) d V = \int_{0}^{r} 4 π r^{2} (1 - 2 m / r)^{- 1 / 2} (ρ - μ_{0} n) d r \end{matrix}

Subtract these from the total mass-energy,

m

; the quantity that is left must be the gravitational potential energy,

\begin{aligned} Ω & = - \int_{0}^{r} ρ [(1 - 2 m / r)^{- 1 / 2} - 1] 4 π r^{2} d r \\ (7) & \approx - \int_{0}^{r} (ρ m / r) 4 π r^{2} d r \\ [Newtonian limit, m / r ≪ 1] \end{aligned}

(See exercise 23.7.)

Turn next to the

\hat{r} \hat{r}

component of the field equations:

\begin{aligned} G_{\hat{r} \hat{r}} & = - r^{- 2} + r^{- 2} e^{- 2 Λ} + 2 r^{- 1} e^{- 2 Λ} d Φ / d r \\ = 8 π T_{\hat{r} \hat{r}} = 8 π p . \end{aligned}

Solving this equation for the derivative of

Φ

, and replacing

e^{- 2 Λ}

1 - 2 m / r

, one obtains an expression for the gradient of the potential

Φ

\begin{matrix} (23.21) & \frac{d Φ}{d r} = \frac{m + 4 π r^{3} p}{r (r - 2 m)} . \end{matrix}

This expression reduces to the familiar formula

\begin{matrix} (23.21~N) & d Φ / d r = m / r^{2} \end{matrix}

in the Newtonian limit.

In most studies of stellar structure, one replaces equation (23.17) by the equivalent equation obtained with the help of (23.21),

\begin{matrix} (23.22) & \frac{d p}{d r} = - \frac{(ρ + p) (m + 4 π r^{3} p)}{r (r - 2 m)} \end{matrix}

This is called the Oppenheimer-Volkoff

(OV)

equation of hydrostatic equilibrium. Its Newtonian limit,

\begin{matrix} (23.22~N) & d p / d r = - ρ m / r^{2} \end{matrix}

is familiar.
Compare two stellar models, one relativistic and the other Newtonian. Suppose that at a given radius

r

[determined in both cases by (proper area)

= 4 π r^{2}

], the two configurations have the same values of

ρ, p

, and

m

. Then in the relativistic model the pressure gradient is

\begin{aligned} \frac{d p}{d (proper radial distance)} & = \frac{d p}{e^{A} d r} \\ (23.23) & = - \frac{(ρ + p) (m + 4 π r^{3} p)}{r^{2} (1 - 2 m / r)^{1 / 2}} . \end{aligned}

In contrast, Newtonian theory gives for the pressure gradient

\begin{matrix} (23.23~N) & \frac{d p}{d (proper radial distance)} = \frac{d p}{d r} = - \frac{ρ m}{r^{2}} \end{matrix}

The relativistic expression for the gradient is larger than the Newtonian expression (1) because the numerator is larger (added pressure term in both factors) and (2) because the denominator is smaller [shrinkage factor

(1 - 2 m / r)^{1 / 2}

]. Therefore, as one proceeds deeper into the star, one finds pressure rising faster than Newtonian gravitation theory would predict. Moreover, this rise in pressure is in a certain sense "self-regenerative." The more the pressure goes up, the larger the pressure-correction terms become in the numerator of (23.23); and the larger these terms become, the faster is the further rise of the pressure as one probes still deeper into the star. The geometric factor

[1 - 2 m (r) / r]^{1 / 2}

in the denominator of (23.23) further augments this regenerative rise of pressure towards the center. It is appropriate to summarize the situation in short-hand terms by saying that general relativity predicts stronger gravitational forces in a stationary body than does Newtonian theory. These forces, among their other important effects, can pull certain white-dwarf stars and supermassive stars into gravitational collapse under circumstances (see Chapter 24) where Newtonian theory would have predicted stable hydrostatic equilibrium. As the most elementary indication that a new factor has surfaced in the analysis of stability, note that no star in hydrostatic equilibrium can ever have

2 m (r) / r \geq 1

(see Box 23.2 for one illustration and

§ 23.8

for discussion), a phenomenon alien to Newtonian theory.
Now in hand are five equations of structure [two equations of state (23.16); equation (23.19), expressing

m (r) = \frac{1}{2} r (1 - e^{- 2 Λ})

as a volume integral of

ρ

; the source

Equation of hydrostatic equilibrium rewritten in "OV" form

Comparison of pressure gradients in Newtonian and relativistic stars

Equations of stellar structure summarized
equation (23.21) for

Φ

; and the OV equation of hydrostatic equilibrium (23.22)] for the five structure functions

ρ, p, n, Φ, Λ

. If the theory of relativistic stars as outlined above is well posed, then each of the remaining eight Einstein field equations

G_{\hat{α} \hat{β}} = 8 π T_{\hat{α} \hat{β}}

must be either vacuous ("

0 = 0

"), or must be a consequence of the five equations of structure. This is, indeed, the case, as one can verify by straightforward but tedious computations.
To construct a stellar model, one needs boundary conditions as well as structure equations. To facilitate the presentation of boundary conditions, the next section will examine the star's external gravitational field.

EXERCISES

Exercise 23.3. LAW OF LOCAL ENERGY-MOMENTUM CONSERVATION (for readers who have not studied Chapter 22)
Evaluate the four components of the equation

T^{α β}_{; β} = 0

for the stress-energy tensor (23.14) in the Schwarzschild coordinate system of equation (23.7). [Answer: only

T^{τ β}_{; β} = 0

gives a nonvacuous result; it gives equation (23.17).]

Exercise 23.4. EINSTEIN CURVATURE TENSOR

(for readers who have not studied Chapter 14)

Calculate the components of the Einstein curvature tensor,

G_{α β}

, in Schwarzschild coordinates. Then perform a transformation to obtain

G_{\hat{α} \hat{β}}

, the components in the orthonormal frame of equations (23.15a,b). [See Box 8.6, or Box 14.2 and equation (14.7).]

Exercise 23.5. TOTAL NUMBER OF BARYONS IN A STAR

Show that, if

r = R

is the location of the surface of a static star, then the total number of baryons inside the star is

\begin{matrix} (23.24) & A = \int_{0}^{R} 4 π r^{2} n e^{A} d r \end{matrix}

[Hint: See the discussion of

m_{0}

in Box 23.1.]

Exercise 23.6. BUOYANT FORCE IN A STAR

An observer at rest at some point inside a relativistic star measures the radial pressure-buoyant force,

F_{buoy}

, on a small fluid element of volume

V

. Let him use the usual laboratory techniques. Do not confuse him by telling him he is in a relativistic star. What value will he find for

F_{buoy}

, in terms of

ρ, p, m, V

, and

d p / d r

? If he equates this buoyant force to an equal and opposite gravitational force,

F_{grav}

, what will

F_{grav}

be in terms of

ρ, p, m, V

, and

r

? (Use equation 23.22.) How do these results differ from the corresponding Newtonian results?

Exercise 23.7. GRAVITATIONAL ENERGY OF A NEWTONIAN STAR

Calculate in Newtonian theory the energy one would gain from gravity if one were to construct a star by adding one spherical shell of matter on top of another, working from the inside outward. Use Laplace's equation

{(r^{2} Φ_{, r})}_{, r} = 4 π r^{2} ρ

and the equation of hydrostatic equilibrium

p_{, r} = - ρ Φ_{, r}

to put the answer in the following equivalent forms:
(energy gained from gravity)

\equiv -

(gravitational potential energy)

\begin{aligned} = \int_{0}^{R} (ρ r Φ_{, r}) 4 π r^{2} d r = \int_{0}^{R} (ρ m / r) 4 π r^{2} d r \\ = - \frac{1}{2} \int_{0}^{R} (ρ Φ) 4 π r^{2} d r = \frac{1}{8 π} \int_{0}^{\infty} {(Φ_{, r})}^{2} 4 π r^{2} d r \\ = 3 \int_{0}^{R} 4 π r^{2} p d r . \end{aligned}

§23.6. EXTERNAL GRAVITATIONAL FIELD

Outside a star the density and pressure vanish, so only the metric parameters

Φ

and

Λ = - \frac{1}{2} \ln (1 - 2 m / r)

need be considered. From equation (23.19) one sees that "the mass inside radius

r

, "

m (r)

, stays constant for values of

r

greater than

R

(outside the star). Its constant value is denoted by

M

\begin{matrix} (23.25) & m (r) = M for r > R (i.e., outside the star). \end{matrix}

By integrating equation (23.21) with

p = 0

and

m = M

, and by imposing the boundary condition (23.10) on

Φ

r = \infty

("normalization of scale of time at

r = \infty

"), one finds

\begin{matrix} (23.26) & Φ (r) = \frac{1}{2} \ln (1 - 2 M / r) for r > R . \end{matrix}

Consequently, outside the star the spacetime geometry (23.7) becomes

\begin{matrix} (23.27) & d s^{2} = - (1 - \frac{2 M}{r}) d t^{2} + \frac{d r^{2}}{(1 - 2 M / r)} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

This is called the "Schwarzschild geometry" or "Schwarzschild gratitational field" or "Schwarzschild line element," because Karl Schwarzschild (1916a) discovered it as an exact solution to Einstein's field equations a few months after Einstein formulated general relativity theory.

In that region of spacetime,

r ≫ 2 M

, where the geometry is nearly flat, Newton's theory of gravity is valid, and the Newtonian potential is

\begin{matrix} (23.26~N) & Φ = - M / r for r > R, r ≫ 2 M . \end{matrix}

Consequently,

M

is the mass that governs the Keplerian motions of planets in the distant, Newtonian gravitational field-i.e., it is the star's "total mass-energy" (see Chapters 19 and 20). Since the metric (23.27) far outside the star is precisely diagonal

(g_{t j} \equiv 0)

, the star's total angular momentum must vanish. This result accords with the absence of internal fluid motions.

Spacetime outside star possesses "Schwarzschild" geometry

§23.7. HOW TO CONSTRUCT A STELLAR MODEL

Equations of stellar structure collected together

The equations of stellar structure (23.16), (23.19), (23.21), (23.22), and associated boundary conditions (to be discussed below), all gathered together along with the line element, read as follows.

Line Element

\begin{aligned} (') & d s^{2} & = - e^{2 ϕ} d t^{2} + \frac{d r^{2}}{1 - 2 m / r} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \\ = - (1 - \frac{2 M}{r}) d t^{2} + \frac{d r^{2}}{1 - 2 M / r} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) for r > R . \end{aligned}

Mass Equation

\begin{matrix} (23.28a) & m = \int_{0}^{r} 4 π r^{2} ρ d r, with m (r = 0) = 0 \end{matrix}

OV Equation of Hydrostatic Equilibrium

\begin{matrix} (23.28b) & \frac{d p}{d r} = - \frac{(ρ + p) (m + 4 π r^{3} p)}{r (r - 2 m)}, with p (r = 0) = p_{c} = central pressure. \end{matrix}

Equations of State

\begin{aligned} (23.28c) & p = p (n), \\ (23.28d) & ρ = ρ (n) . \end{aligned}

Source Equation for $Φ$

\begin{matrix} (23.28e) & \frac{d Φ}{d r} = \frac{(m + 4 π r^{3} p)}{r (r - 2 m)}, with Φ (r = R) = \frac{1}{2} \ln (1 - 2 M / R) . \end{matrix}

To construct a stellar model one can proceed as follows. First specify the equations of state

(23.28 c, d)

and a value of the central pressure,

p_{c}

. Also specify an arbitrary (later to be renormalized) value,

Φ_{0}

, for

Φ (r = 0)

. The boundary conditions

p (r = 0) = p_{c}, Φ (r = 0) = Φ_{0}, m (r = 0) = 0

are sufficient to determine uniquely the solution to the coupled equations (23.28). Integrate these coupled equations outward from

r = 0

until the pressure vanishes. [The OV equation, (23.28b), guarantees that the pressure will decrease monotonically so long as the equations of state obey the
reasonable restriction

ρ \geq 0

for all

p \geq 0

.] The point at which the pressure reaches zero is the star's surface; the value of

r

there is the star's radius,

R

; and the value of

m

there is the star's total mass-energy,

M

. Having reached the surface, renormalize

Φ

by adding a constant to it everywhere, so that it obeys the boundary condition (23.28e). The result is a relativistic stellar model whose structure functions

Φ, m

ρ, p, n

satisfy the equations of structure.

Notice that for any fixed choice of the equations of state

p = p (n), ρ = ρ (n)

, the stellar models form a one-parameter sequence (parameter

p_{c}

). Once the central pressure has been specified, the model is determined uniquely.
The next chapter describes a variety of realistic stellar models constructed numerically by the above prescription. For an idealized stellar model constructed analytically, see Box 23.2.

Exercise 23.8. NEWTONIAN STARS OF UNIFORM DENSITY

EXERCISE

Calculate the structures of uniform-density configurations in Newtonian theory. Show that the relativistic configurations of Box 23.2 become identical to the Newtonian configurations in the weak-gravity limit. Also show that there are no mass or radius limits in Newtonian theory.
(continued on page 612)

Box 23.2 RELATIVISTIC MODEL STAR OF UNIFORM DENSITY

For realistic equations of state (see next chapter), the equations of stellar structure (23.28) cannot be integrated analytically; numerical integration is necessary. However, analytic solutions exist for various idealized and ad hoc equations of state. One of the most useful analytic solutions [Karl Schwarzschild (1916b)] describes a star of uniform density,

\begin{matrix} (1) & ρ = ρ_{0} = constant for all p . \end{matrix}

It is not necessary to indulge in the fiction of "an incompressible fluid" to accept this model as interesting. Incompressibility would imply a speed of sound,

v = (d p / d ρ)^{1 / 2}

, of unlimited magnitude, therefore in excess of the speed of light, and therefore in contradiction with a central principle of special relativity ("principle of causality") that no physical effect can be propagated at a speed

v > 1

. (If a source could cause an effect so quickly in one local Lorentz frame, then there would exist another local Lorentz frame in which the effect would occur before the source had acted!) However, that the part of the fluid in the region of high pressure has the same density as the part of the fluid in the region of low pressure is an idea easy to admit, if only one thinks of the fluid having a composition that varies from one

Box 23.2 (continued)

r

value to another ("hand-tailored"). Whether one thinks along this line, or simply has in mind a globe of water limited in size to a small fraction of the dimensions of the earth, one has in Schwarzschild's model an instructive example of hydrostatics done in the framework of Einstein's theory.

The mass equation (23.28a) gives immediately

\begin{matrix} (2) & m = {\begin{cases} (4 π / 3) ρ_{0^{'}} r^{3} & for r < R \\ M = (4 π / 3) ρ_{0} R^{3} & for r > R \end{cases}} . \end{matrix}

from which follows the length-correction factor in the metric

\begin{matrix} (3) & \frac{d (proper distance)}{d r} = e^{Λ} = [1 - 2 m (r) / r]^{- 1 / 2} . \end{matrix}

When for ease of visualization the space geometry

(r, ϕ)

of an equatorial slice through the star is viewed as embedded in a Euclidean 3-geometry

(z, r, ϕ)

[see

$ 23.8]

, the "lift" out of the plane

z = 0

\begin{aligned} z (r) = {\begin{matrix} {(R^{3} / 2 M)}^{1 / 2} [1 - {(1 - 2 M r^{2} / R^{3})}^{1 / 2}] for r \leq R, \\ {(R^{3} / 2 M)}^{1 / 2} [1 - (1 - 2 M / R)^{1 / 2}] + [8 M (r - 2 M)]^{1 / 2} - [8 M (R - 2 M)]^{1 / 2} \end{matrix} \\ (4) & for r \geq R . \end{aligned}

The knowledge of

m (r)

from (2) allows the equation of hydrostatic equilibrium ( 23.28 b ) to be integrated to give the pressure:

\begin{matrix} (5) & p = ρ_{0} {\frac{{(1 - 2 M r^{2} / R^{3})}^{1 / 2} - (1 - 2 M / R)^{1 / 2}}{3 (1 - 2 M / R)^{1 / 2} - {(1 - 2 M r^{2} / R^{3})}^{1 / 2}}} for r < R . \end{matrix}

The pressure in turn leads via (23.28e) to the time-correction factor in the metric.

\frac{d (proper time)}{d t} = e^{ϕ} = {\begin{cases} \frac{3}{2} {(1 - \frac{2 M}{R})}^{1 / 2} - \frac{1}{2} {(1 - \frac{2 M r^{2}}{R^{3}})}^{1 / 2} & for r < R \\ (1 - 2 M / r)^{1 / 2} & for r > R \end{cases}}

.
Several features of these uniform-density configurations are noteworthy. (1) For fixed energy density,

ρ_{0}

, the central pressure

\begin{matrix} (7) & p_{c} = ρ_{0} {\frac{1 - (1 - 2 M / R)^{1 / 2}}{3 (1 - 2 M / R)^{1 / 2} - 1}}, \end{matrix}

increases monotonically as the radius,

R

, increases-and, hence, also as the mass,

M = (4 π / 3) ρ_{0} R^{3}

, and the ratio ("strength of gravity")

\begin{matrix} (8) & 2 M / R = (8 π / 3) ρ_{0} R^{2} \end{matrix}

increase. This is natural, since, as more and more matter is added to the star, a greater and greater pressure is required to support it. (2) The central pressure becomes infinite when

M, R

, and

2 M / R

reach the limiting values

\begin{aligned} (9) & R_{lim} = & (9 / 4) M_{lim} = {(3 π ρ_{0})}^{- 1 / 2}, \\ (10) & (2 M / R)_{lim} = 8 / 9 . \end{aligned}

No star of uniform density can have a mass and radius exceeding these limits. These limits are purely relativistic phenomena; no such limits occur in Newtonian theory. (3) Inside the star the space geometry (geometry of a hypersurface

t =

constant) is that of a three-dimensional spherical surface with radius of curvature

\begin{matrix} (11) & a = {(3 / 8 π ρ_{0})}^{1 / 2} . \end{matrix}

[See equation (4), above.] Outside the star the (Schwarzschild) space geometry is that of a three-dimensional paraboloid of revolution. The interior and exterior geometries join together smoothly. All these details are shown in the following three diagrams. There all quantities are given in the following geometric units (to convert mass in g or density in

g / {cm}^{3}

into mass in cm or density in

{cm}^{- 2}

, multiply by

0.742 \times 10^{- 28} cm / g)

: lengths, in units

{(3 / 8 π ρ_{0})}^{1 / 2}

; pressure, in units

ρ_{0}

; mass, in units

{(3 / 32 π ρ_{0})}^{1 / 2}

Box 23.2 (continued)

The mass "after assembly" is what is called

M

. The mass of the same fluid, dispersed in droplets at infinite separation, is called

M_{before}

in the following table.

M_{before}

small

0.0882

0.894

1.0913

1.374

M

small

0.0828

0.636

0.729

0.838 (critical)

Difference

(binding):

\frac{3}{10} M^{5 / 3}

0.0054

0.258

0.362

0.536

§23.8. THE SPACETIME GEOMETRY FOR A STATIC STAR

Surface area of spheres,

4 π r^{2}

:
(1) increases monotonically from center of star outward

For a highly relativistic star, the spacetime geometry departs strongly from EuclidLorentz flatness. Consequently, there is no a priori reason to expect that the surface area

4 π r^{2}

, and hence also the radial coordinate

r

, will increase monotonically as one moves from the center of the star outward. Fortunately, the equations of stellar structure guarantee that r will increase monotonically from 0 at the star's center to

\infty

at an infinite distance away from the star, so long as

ρ \geq 0

and so long as the star is static (equilibrium).
The monotonicity of

r

can be seen as follows. Introduce as a new radial coordinate proper distance,

ℓ

, from the center of the star. By virtue of expression (23.27') for the metric,

ℓ

and

r

are related by

\begin{matrix} (23.29) & d r = \pm (1 - 2 m / r)^{1 / 2} d ℓ . \end{matrix}

Note that

r

is zero at the center of the star (where

m \propto r^{3}

), and note that

r

is always nonnegative by definition. Therefore

r

must at first increase with

ℓ

as one moves outward from

ℓ = 0; r (ℓ)

can later reach a maximum and start decreasing only at a point where

2 m / r

becomes unity [see equation (23.29)]. Such a behavior can and does happen in a closed model universe, a 3 -sphere of uniform density and radius

a

, where

r (ℓ) = a \sin (ℓ / a)

[see Chapter 27; especially the embedding diagram of Box 27.2(A)]. However, the field equations demand that such a system be dynamic. Here, on the contrary, attention is limited to a system where conditions are static. In such a system, the condition of hydrostatic equilibrium (23.28b) applies. Then the pressure gradient is given by an expression with the factor

[1 - 2 m (r) / r]

in its denominator. If

2 m / r

approaches unity with increasing

ℓ

in some region of the star, the pressure gradient
there becomes so large that one comes to the point

p = 0

(surface of the star) before one comes to any point where

2 m (r) / r

might attain unit value. Moreover, after the surface of the star is passed,

m

remains constant,

m (r) = M

, and

2 m (r) / r

decreases. Consequently,

2 m / r

is always less than unity; and

r (ℓ)

cannot have a maximum, Q.e.D. (Details of the proof are left to the reader as exercise 23.9.)

Although the radii of curvature,

r

, and corresponding spherical surface areas,

4 π r^{2}

, increase monotonically from the center of a star outward, they do not increase at the same rate as they would in flat spacetime. In flat spacetime the rate of increase is given by

d r / d (

proper radial distance

) = d r / d ℓ = 1

. In a star it is given by

d r / d ℓ = (1 - 2 m / r)^{1 / 2} < 1

. Consequently, if one were to climb a long ladder outward from the center of a relativistic star, measuring for each successive spherical shell its Schwarzschild

r

-value ("proper circumference"

/ 2 π

), one would find these

r

-values to increase surprisingly slowly.

This strange behavior is most easily visualized by means of an "embedding diagram." It would be too much for any easy visualization if one were to attempt to embed the whole curved four-dimensional manifold in some higher-dimensional flat space. [See, however, Fronsdal (1959) and Clarke (1970) for a global embedding in

5 + 1

dimensions, and Kasner (1921b) for a local embedding in

4 + 2

dimensions. One can never embed a non-flat, vacuum metric

(G_{μ ν} = 0)

in a flat space of 5 dimensions (Kasner, 1921c).] Therefore seek a simpler picture (Flamm 1916). Space at one time in the context of a static system has the same 3-geometry as space at another time. Therefore, depict 3 -space only as it is at one time,

t =

constant. Moreover, at any one time the space itself has spherical symmetry. Consequently, one slice through the center,

r = 0

, that divides the space symmetrically into two halves (for example, the equatorial slice,

θ = π / 2

) has the same 2 -geometry as any other such slice (any selected angle of tilt, at any azimuth) through the center. Therefore limit attention to the 2 -geometry of the equatorial slice. The geometry on this slice is described by the line element

\begin{matrix} (23.30) & d s^{2} = [1 - 2 m (r) / r]^{- 1} d r^{2} + r^{2} d ϕ^{2} \end{matrix}

Now one may embed this two-dimensional curved-space geometry in the flat geometry of a Euclidean three-dimensional manifold.

If the curvature of the two-dimensional slice is zero or negligible, the embedding is trivial. In this event, identify the 2-geometry with the slice

z = 0

of the Euclidean 3 -space. Moreover, introduce into that 3 -space the familiar cylindrical coordinates

z, r, ϕ

, that one employs for any problem with axial symmetry (see Fig. 23.1 and Box 23.2 for more detail). Then one recognizes the flat two-dimensional slice as the set of points of the Euclidean space with

z = 0

, with

ϕ

running from 0 to

2 π

, and

r

from 0 to

\infty

. One has identified the

r

and

ϕ

of the slice with the

r

and

ϕ

of the Euclidean 3-space.

If the 2-geometry is curved, as it is when the equatorial section is taken through a real star, then maintain the identification between the

r, ϕ

, of the slice and the

r, ϕ

, of the Euclidean 3-geometry, but bend up the slice out of the plane

z = 0

(except at the origin,

r = 0

). At the same time, insist that the bending be axially symmetric. In other words, require that the amount of the "lift" above the plane

z = 0

shall
(2) but increases more slowly than in flat spacetime

Embedding of spacetime in a flat space of higher dimensionality

Construction of "embedding diagram" for equatorial slice through star

Figure 23.1.
Geometry within (grey) and around (white) a star of radius

R = 2.66 M

, schematically displayed. The star is in hydrostatic equilibrium and has zero angular momentum (spherical symmetry). The twodimensional geometry

d s^{2} = [1 - 2 m (r) / r]^{- 1} d r^{2} + r^{2} d ϕ^{2}

of an equatorial slice through the star

(θ = π / 2, t =

constant

)

is represented as embedded in Euclidean 3 -space, in such a way that distances between any two nearby points

(r, ϕ)

and (

r + d r, ϕ + d ϕ

) are correctly reproduced. Distances measured off the curved surface have no physical meaning; points off that surface have no physical meaning; and the Euclidean 3 -space itself has no physical meaning. Only the curved 2-geometry has meaning. A circle of Schwarzschild coordinate radius

r

has proper circumference

2 π r

(attention limited to equatorial plane of star,

θ = π / 2

). Replace this circle by a sphere of proper area

4 π r^{2}

, similarly for all the other circles, in order to visualize the entire 3-geometry in and around the star at any chosen moment of Schwarzschild coordinate time

t

. The factor

[1 - 2 m (r) / r]^{- 1}

develops no singularity as

r

decreases within

r = 2 M

, because

m (r)

decreases sufficiently fast with decreasing

r

.
be independent of

ϕ

, whatever may be its dependence on

r

. Thus the whole story of the embedding is summarized by the single function, the lift,

z = z (r) ("embedding formula").

The geometry on this curved two-dimensional locus in Euclidean space (a made-up 3 -space; it has nothing whatever to do with the real world) is to be identical with the geometry of the two-dimensional equatorial slice through the actual star; in other words, the line elements in the two cases are to be identical. To work out this requirement in mathematical terms, write the line element in three-dimensional Euclidean space in the form

\begin{matrix} (23.31) & d s^{2} = d z^{2} + d r^{2} + r^{2} d ϕ^{2} \end{matrix}

Restrict to the chosen locus ("lifted surface") by writing

z = z (r)

d z = (d z / d r) d r

. Thus have

\begin{matrix} (23.32) & d s^{2} = [1 + {(\frac{d z (r)}{d r})}^{2}] d r^{2} + r^{2} d ϕ^{2} \end{matrix}

on the two-dimensional locus in the 3-geometry, to be identified with

d s^{2} = [1 - 2 m (r) / r]^{- 1} d r^{2} + r^{2} d ϕ^{2}

in the actual star. Compare and conclude

\begin{matrix} (23.33) & {(\frac{d z (r)}{d r})}^{2} + 1 = [1 - 2 m (r) / r]^{- 1} \end{matrix}

This equation is information enough to find the lift as a function of

r

; thus,

\begin{matrix} (23.34a) & \begin{array}{cc} z (r) = \int_{0}^{r} \frac{d r}{{[\frac{r}{2 m (r)} - 1]}^{1 / 2}} & everywhere, \\ z (r) = [8 M (r - 2 M)]^{1 / 2} + constant & outside the star. \end{array} \end{matrix}

Outside the star this embedded surface is a segment of a paraboloid of revolution. Its form inside the star depends on how the mass,

m

, varies as a function of

r

. Recall

Description of embedded surface that

m (r)

varies as

(4 π / 3) ρ_{c} r^{3}

near the center of the star. Conclude that the embedded surface there looks like a segment of a sphere of radius

a = {(3 / 8 π ρ_{c})}^{1 / 2}

; thus,

\begin{matrix} (23.34c) & [a - z (r)]^{2} + r^{2} = a^{2} for r ≪ a = {(3 / 8 π ρ_{c})}^{1 / 2} \end{matrix}

In the special case of a star with uniform density (Box 23.2), the entire interior is of the spherical form (23.34c); in the general case it is not. In all cases, because

r > 2 m (r)

, equation (23.34a) produces a surface with

z

and

r

as monotonically increasing functions of each other. This means that the embedded surface always opens upward and outward like a bowl; it always looks qualitatively like Figure 23.1; it never has a neck, and it never flattens out except asymptotically at

r = \infty

. At the star's surface, even though the density may drop discontinuously to zero (

ρ

finite inside when

p = 0; ρ

zero outside), the interior and exterior geometries will join together smoothly [

d z / d r

, as given by equation (23.33), is continuous].

It must be emphasized that only points lying on the embedded 2 -surface have physical significance so far as the stellar geometry is concerned: the three-dimensional regions inside and outside the bowl of Figure 23.1 are physically meaningless. So is the Euclidean embedding space. It merely permits one to visualize the geometry of space around the star in a convenient manner.

Exercise 23.9. GOOD BEHAVIOR OF r

EXERCISES

Carry out explicitly the full details of the proof, at the beginning of this section, that

2 m / r

is always less than unity and

r

is a monotonic function of

ℓ

Exercise 23.10. CENTER OF STAR OCCUPIED BY IDEAL FERMI GAS AT EXTREME RELATIVISTIC LIMIT

Opposite to the idealization of a star built from an incompressible fluid is the idealization in which it is built from an ideal Fermi gas [ideal neutron star; see Oppenheimer and Volkoff (1939)] at zero temperature, so highly compressed that the particles have relativistic energies,
in comparison with which any rest mass they possess is negligible. In this limit, with two particles per occupied cell of volume

h^{3}

in phase space, one has

\begin{aligned} (\binom{number density}{of fermions}) = n = (2 / h^{3}) 4 π \int_{0}^{p_{F}} p^{2} d p = 8 π p_{F}^{3} / 3 h^{3}, \\ (\binom{density of}{mass-energy}) = ρ = (2 / h^{3}) 4 π \int_{0}^{p_{F}} c p \cdot p^{2} d p = 2 π c p_{F}^{4} / h^{3}, \end{aligned}

and finally

p = - \frac{d (\binom{energy}{per particle})}{d (\binom{volume}{per particle})} = - \frac{d (ρ / n)}{d (1 / n)} = 2 π c p_{F}^{4} / 3 h^{3} = ρ / 3

as if one were dealing with radiation instead of particles (

p_{F} =

Fermi momentum; momentum of highest occupied state).

Box 23.3 RIGOROUS DERIVATION OF THE SPHERICALLY SYMMETRIC LINE ELEMENT

Section 23.2 gave a heuristic derivation of the general spherically symmetric line element (23.7). This box attempts a more rigorous derivation, applicable to nonstatic systems, as well as static ones.

Begin with a manifold

M^{4}

on which a metric

d s^{2}

of Lorentz signature is defined. Assume

M^{4}

to be spherically symmetric in the sense that to any

3 \times 3

rotation matrix

A

there corresponds a mapping (rotation) of

M^{4}

, also called

A (A : M^{4}

⟶ M^{4} : P ⟶ A P

), that preserves the lengths of all curves. Further assumptions and constructions will be numbered (i), (ii), etc., so one can see what specializations are needed to get to the line element (23.7). Daggers (

†

) indicate assumptions that are found inapplicable to some other physically interesting situations.

For any point

P

, form the set

s = S (P) =

{A P \in M^{4} ∣ A \in S O (3)}

of all points equivalent to

P

under rotations. Assume (i)

†

that

s

is a two-dimensional surface (except for center points, where

s

is zero-dimensional), and (ii) that the metric on

s

is that of a standard 2 -sphere. Then on

s

one will have

\begin{matrix} (1) & {(d s^{2})}_{s} = R^{2} (s) d Ω^{2} \end{matrix}

where

d Ω^{2}

is the standard metric of a unit sphere

(d Ω^{2} = d θ^{2} + \sin^{2} θ d ϕ^{2}

for some

θ, ϕ

, defined on

s)

, and where

2 π R

is the circumference of

s

. If

M^{2}

is the set of all such surfaces

s

, then

S : M^{4} ⟶

M^{2} : P ⟶ s = S (P)

allows one to obtain, from

R : M^{2} ⟶ R : s ⟶ R (s)

[the "circumference" function on

M^{2}

as defined by equation (1)], a corresponding function

R : M^{4} ⟶ ℜ : P ⟶

R (S (P))

M^{4}

which in some cases can eventually be used as a coordinate on

M^{4}

. (Note:

Ω

denotes here the real numbers.)

Now assume (iii)

†

there is a spherically symmetric 4 -velocity field

u

, defined so that if

P = C (τ)

is one trajectory of

u

with

u = d / d τ

, then each curve

P = A C (τ)

obtained by a rotation must also be a trajectory of

u

. The orthogonal projection of

u

onto any sphere

s

must then vanish, as there are no rotation invariant non-zero vector fields on 2 -spheres. Thus

u

is orthogonal to each

s

. Also, if two trajectories of

u

start on some same sphere

s

, so

C_{1} (0) = A C_{2} (0)

, then the same rotation

A

will always relate them,

C_{1} (τ) = A C_{2} (τ)

, since trajectories are uniquely defined by any one point on them. Then

S (C_{1} (τ))

and

S (C_{2} (τ))

are both the same curve in

M^{2}

, whose tangent

d / d τ

one can call also

u

; in this way one obtains a vector field

u

M^{2}

. Give each trajectory of

u

M^{2}

a different label

r

to define a function

r (s)

M^{2}

. Denote by

r = r (S (P))

a corresponding function

r

M^{4}

with

d r / d τ = 0

. Since functions and their gradients on

M^{4}

define corresponding quantities on

M^{2}

, inner products such as

d f \cdot d g

can be defined on

M^{2}

by their values on

M^{4}

; thus, from the metric on

M^{4}

one obtains a metric on

M^{2}

. Then by equa-
(a) Write out the relativistic equation of hydrostatic equilibrium for a substance satisfying the equation of state

p = ρ / 3

.
(b) Show that there exists a well-defined analytic solution for the limiting case of infinite central density, in which

m (r) / r

has the value

3 / 14

.
(c) Find

ρ (r), p (r)

, and

n (r)

.
(d) Show that the number of particles out to any finite

r

-value is finite, despite the fact that

n (r)

is infinite at the origin.
(e) Show that the 3-geometry has a "conical singularity" at

r = 0

.
(f) Make an "embedding diagram" for this 3-geometry ["lift"

z (r)

as a function of

r

from (23.34)]. (Note that the conical singularity at

r = 0

, otherwise physically unreasonable, arises because the density of mass-energy goes to infinity at that point. Note also that the calculated mass of the system diverges to infinity as

r ⟶ \infty

. In actuality with decreasing density the Fermi momentum falls from relativistic to nonrelativistic values, the equation of state changes its mathematical form, and the total mass

M

converges to a finite value).
tion (23.5) or equivalently by drawing curves in

M^{2}

orthogonal to the

r =

const. lines, and giving each a different label

t

, one obtains coordinates with

g^{r t} = d r \cdot d t = 0

. Both

r

and

t

labels were assigned arbitrarily on the corresponding curves, so it is clear that transformations

t^{'} = t^{'} (t)

and

r^{'} = r^{'} (r)

are not excluded.

On one 2-sphere

s

M^{4}

, on the

t = 0

hypersurface, choose a set of

θ, ϕ

coordinates by picking the pole

(θ = 0)

and the prime meridian

(ϕ = 0)

arbitrarily. Then extend the definition of

θ, ϕ

, over the

t = 0

hypersurface by requiring

θ

and

ϕ

to be constant on curves orthogonal to each 2 -sphere

s

, i.e., by demanding that

(\partial / \partial r)_{θ ϕ}

be orthogonal to each

s

t = 0

. Extend the definition of

θ

and

ϕ

t \neq 0

by requiring them to be constant on curves with tangent

u

, so

(\partial / \partial t)_{r θ ϕ} \propto u

. But each

s

is a surface of constant

r

and

t

; so

(\partial / \partial θ)_{r t ϕ}

and

(\partial / \partial ϕ)_{r t θ}

are tangent to

s

, while

u \propto (\partial / \partial t)

is orthogonal to each

s

. Consequently,
and

\begin{matrix} (2) & g_{t θ} = (\partial / \partial t) \cdot (\partial / \partial θ) = 0 \end{matrix}

\begin{matrix} (3) & g_{t ϕ} = (\partial / \partial t) \cdot (\partial / \partial ϕ) = 0 \end{matrix}

in the

t r θ \dot{ϕ}

coordinate system just constructed. The vector

(\partial / \partial r)_{t θ ϕ}

does not depend on the arbitrary directions introduced in the original choice of

θ, ϕ

coordinates on one sphere

s

; it is invariant under transformations

θ = θ (θ^{'}, ϕ^{'}), ϕ = ϕ (θ^{'}, ϕ^{'})

. But nothing except

θ

and

ϕ

introduced nonrotationally invariant elements into the discussion; so

(\partial / \partial r)_{t θ ϕ}

must be a rotationally invariant vector field (un-
like, say,

\partial / \partial ϕ)

; so it is, like

u

, orthogonal to each 2 -sphere

s

. This invariance then gives

\begin{aligned} (4) & g_{r θ} = (\partial / \partial r) \cdot (\partial / \partial θ) = 0, \\ (5) & g_{r ϕ} = (\partial / \partial r) \cdot (\partial / \partial ϕ) = 0, \end{aligned}

which, with

g^{t r} = 0

as previously established, gives

g_{t r} = 0

. The result is a line element of the form (23.3). Further specialization, a change of radial and time coordinates to

R

and

T

, where

R

is defined by (1) above and

\begin{aligned} d T & = e^{ψ} [\frac{1}{g_{r r}} \frac{\partial R}{\partial r} d t - \frac{1}{g_{t t}} \frac{\partial R}{\partial t} d r], \\ e^{ψ} & = (\binom{integrating}{factor}), \end{aligned}

followed by a change of notation, leads to Schwarzschild coordinates and the line element (23.7)-though such a transformation is possible (i.e., nonsingular) only where

d R \land d T \neq 0

(\nabla R)^{2} = \frac{(\partial R / \partial t)^{2}}{g_{t t}} + \frac{(\partial R / \partial r)^{2}}{g_{π r}} \neq 0

If (iv)

†

spacetime is asymptotically flat, so

r ⟶ \infty

is a region where the metric can take on its special relativity values, then the arbitrariness in the

t

coordinate,

t^{'} = t^{'} (t)

, can be eliminated by requiring

g_{t t} = - 1

r ⟶ \infty

. Then

(\partial / \partial t)_{r θ ϕ}

is uniquely determined by natural requirements (independent of the arbitrary

θ, ϕ

, choices), and whenever it is desired to make the further physical assumption (v)

†

of a time-independent geometry, this can be appropriately restated as

\partial g_{μ ν} / \partial t = 0

anme 24

PULSARS AND NEUTRON STARS; QUASARS AND SUPERMASSIVE STARS

Go, wond'rous creature, mount where Science guides, Go, measure earth, weigh air, and state the tides;

Instruct the planets in what orbs to run,
Correct old time, and regulate the sun.

ALEXANDER POPE (1733)

§24.1. OVERVIEW

Types of stellar configurations where relativity should be important

Five kinds of stellar configurations are recognized in which relativistic effects should be significant: white dwarfs, neutron stars, black holes, supermassive stars, and relativistic star clusters. The key facts about each type of configuration are summarized in Box 24.1; and the most important details are described in the text of this chapter (white dwarfs in

§ 24.2

; neutron stars and their connection to pulsars in

§ § 24.2

and 24.3 ; supermassive stars and their possible connection to quasars and galactic nuclei in

§ § 24.4

and 24.5 ; and relativistic star clusters in

§ 24.6

; a detailed discussion of black holes is delayed until Chapter 33).

The book Stars and Relativity by Zel'dovich and Novikov (1971) presents a clear and very complete treatment of all these astrophysical applications of relativistic stellar theory. In a sense, that book can be regarded as a companion volume to this one; it picks up, with astrophysical emphasis, all the topics that this book treats with gravitational emphasis. This chapter is meant only to give the reader a brief survey of the material to be found in Stars and Relativity.
(continued on page 621)

Box 24.1. STELLAR CONFIGURATIONS WHERE RELATIVISTIC EFFECTS ARE IMPORTANT
[For detailed analyses and references on all these topics, see Zel'dovich and Novikov (1971).]

A. White Dwarf Stars

Are stars of about one solar mass, with radii about 5,000 kilometers and densities about

10^{6} g / {cm}^{3} \sim 1

ton

/ {cm}^{3}

; support themselves against gravity by the pressure of degenerate electrons; have stopped burning nuclear fuel, and are gradually cooling as they radiate away their remaining store of thermal energy.
Were observed and studied astronomically long before they were understood theoretically.
Key points in history:
August 1926, Dirac (1926) formulated FermiDirac statistics, following Fermi (February). December 1926, R. H. Fowler (1926) used Fermi-Dirac statistics to explain the nature of white dwarfs; he invoked electron degeneracy pressure to hold the star out against the inward pull of gravity.
1930, S. Chandrasekhar (1931a,b) calculated white-dwarf models taking account of special relativistic effects in the electron-degeneracy equation of state; he discovered that no white dwarf can be more massive than

\sim 1.2

solar masses ("Chandrasekhar Limit").
1932, L. D. Landau (1932) gave an elementary explanation of the Chandrasekhar limit.
1949, S. A. Kaplan (1949) derived the effects of general relativity on the mass-radius curve for massive white dwarfs, and deduced that general relativity probably induces an instability when the radius becomes smaller than

1.1 \times 10^{3} km

.
Role of general relativity in white dwarfs:
negligible influence on structure;
significant influence on stability, on pulsation
frequencies, and on form of mass-radius curve near the Chandrasekhar limit (i.e., in massive white dwarfs). Electron capture also significant. See, e.g., Zel'dovich and Novikov (1971); Faulkner and Gribbin (1968).

B. Neutron Stars

Are stars of about one solar mass, with radii about 10 km and densities about

10^{14} g / {cm}^{3}

(same as density of an atomic nucleus); are supported against gravity by the pressure of degenerate neutrons and by nucleon-nucleon strong-interaction forces; are not burning nuclear fuel; the energy being radiated is the energy of rotation and the remaining store of internal thermal energy.
Theoretical calculations predicted their existence in 1934, but they were not verified to exist observationally until 1968.
Key points in history:
1932, neutron discovered by Chadwick (1932). 1933-34, Baade and Zwicky (1934a,b,c) (1) invented the concept of neutron star; (2) identified a new class of astronomical objects which they called "supernovae"; (3) suggested that supernovae might be created by the collapse of a normal star to form a neutron star. (See Figure 24.1.)
1939, Oppenheimer and Volkoff (1939) performed the first detailed calculations of the structures of neutron stars; in the process, they laid the foundations of the general relativistic theory of stellar structure as presented in Chapter 23. (See Figure 24.1.)
1942, Duyvendak (1942) and Mayall and Oort (1942) deduced that the Crab nebula is a remnant of the supernova observed by Chi-

Box 24.1 (continued)
nese astronomers in A.D. 1054. Baade (1942) and Minkowskii (1942) identified the "south preceding star," near the center of the Crab Nebula, as probably the (collapsed) remnant of the star that exploded in 1054 (see frontispiece).
1967, Pulsars were discovered by Hewish et al. (1968).
1968, Gold (1968) advanced the idea that pulsars are rotating neutron stars; and subsequent observations confirmed this suggestion.
1969, Cocke, Disney, and Taylor (1969) discovered that the "south preceding star" of the Crab nebula is a pulsar, thereby clinching the connection between supernovae, neutron stars, and pulsars.
Role of general relativity in neutron stars:
significant effects (as much as a factor of 2 ) on structure and vibration periods;
gravitational radiation reaction may be the dominant force that damps nonradial vibrations.

C. Black Holes

Are objects created when a star collapses to a size smaller than twice its geometrized mass

(R < 2 M \sim (M / M_{⊙}) \times 3 km)

, thereby creating such strong spacetime curvatures that it can no longer communicate with the external universe (detailed analysis of black holes in Chapters 33 and 34).
No one who accepts general relativity has found any way to escape the prediction that black holes must exist in our galaxy. This prediction depends in no way on the complexity of the collapse that forms the black holes, or on unknown properties of matter at high density. However, the existence of black holes has not yet been verified observationally.

Key points in history:
1795, Laplace (1795) noted that, according to Newtonian gravity and Newton's corpuscular theory of light, light cannot escape from a sufficiently massive object (Figure 24.1).
1939, Oppenheimer and Snyder (1939) calculated the collapse of a homogeneous sphere of pressure-free fluid, using general relativity, and discovered that the sphere cuts itself off from communication with the rest of the universe. This was the first calculation of how a black hole can form (Figure 24.1).

1965, Beginning of an era of intensive theoretical investigation of black-hole physics. Role of general relativity in black-hole physics: No sensible account of black holes possible in Newtonian theory. The physics of black holes calls on Einstein's description of gravity from beginning to end.

D. Supermassive Stars

Are stars of mass between

10^{3}

and

10^{9}

solar masses, constructed from a hot plasma of density typically less than that in normal stars; are supported primarily by the pressure of photons, which are trapped in the plasma and are in thermal equilibrium with it; burn nuclear fuel (hydrogen) at some stages in their evolution.
Theoretical calculations suggest (but not with complete confidence) that supermassive stars exist in the centers of galaxies and quasars, and perhaps elsewhere. Supermassive stars conceivably could be the energy sources for some quasars and galactic nuclei. However, astronomical observations have not yet yielded definitive evidence about their existence or their roles in the universe if they do exist.

Key points in history:
1963, Hoyle and Fowler (1963a,b) conceived the idea of supermassive stars, calculated their properties, and suggested that they might be associated with galactic nuclei and quasars.
1963-64, Chandrasekhar (1964a,b) and Feynman (1964) developed the general relativistic theory of stellar pulsations; and Feynman used it to show that supermassive stars, although Newtonian in structure, are subject to a general-relativistic instability.
1964 and after, calculations by many workers have elaborated on and extended the ideas of Hoyle and Fowler, but have not produced any spectacular breakthrough.
Role of general relativity in supermassive stars: negligible influence on structure, except in the extreme case of a compact, rapidly rotating, disc-like configuration [see Bardeen and Wagoner (1971); Salpeter and Wagoner (1971)].
significant influence on stability.

E. Relativistic Star Clusters

Are clusters of stars so dense that relativistic corrections to Newtonian theory modify their structure.

Theoretical calculations suggest that relativistic star clusters might, but quite possibly do not, form in the nuclei of some galaxies and quasars; if they do try to form, they might be destroyed during formation by star-star collisions, which convert the cluster into supermassive stars or into a dense conglomerate of stars and gas. Astronomical observations have yielded no definitive evidence, as yet, about the existence of relativistic clusters.
Key points in history:
1965, Zel'dovich and Podurets (1965) conceived the idea of relativistic star clusters, developed the theory of their structure using general relativity and kinetic theory (cf.

$ 25.7

), and speculated about their stability.
1968, Ipser (1969) developed the theory of star-cluster stability and showed (in agreement with the Zel'dovich-Podurets speculations) that, when it becomes too dense, a cluster begins to collapse to form a black hole.
Role of general relativity in star clusters:
significant effect on structure when gravitational redshift from center to infinity exceeds

z_{c} \equiv Δ λ / λ \sim 0.05

.
induces collapse of cluster to form black hole when central redshift reaches

z_{c} \approx 0.50

§24.2. THE ENDPOINT OF STELLAR EVOLUTION

After the normal stages of evolution, stars "die" by a variety of processes. Some stars explode, scattering themselves into the interstellar medium; others contract into a white-dwarf state; and others-according to current theory-collapse to a neutron-star state, or beyond, into a black hole. Although one knows little at present about a star's dynamic evolution into its final state, much is known about the final states themselves. The final states include dispersed nebulae, which are of no interest here; cold stellar configurations, the subject of this section; and "black holes," the subject of Part VII.

Minutes of the Stanford Meeting, Decemper 15-16, 1933

Supernovae and Cosmic Rays. W. BaAde, Mt. Wilson Observatory, aND F. flare up in every stellar system of Technology. - Supernovae flare up ine lifetime of a super(nebula) once in severy days and its absolute brightness at nova is about $\begin{aligned} maximum may be as high as M_{vis} = - 14^{M} . The visible \\ matimes the \end{aligned}$ madiation $L_{ν}$ of a supernova is about $10^{8}$ times the radiation of our sun, that is, $L_{ν} = 3.78 \times 10^{4}$ visible and invisible, is indicate that the total radiation, $^{48} ergs / \sec$ . The superof the order $L_{τ} = 10^{7} L_{ν} = 3.18 \times$ its life a total energy
quite ordinary stars of mass $M < 10^{34} g, E_{τ} / c^{2}$ is of the same order as $M$ itself. In the supernovapothesis suggests sulk is annihilated. In addition the supernovae. Assuming itself that cosmic rays are producena occurs every thousand that in every nebula one supernovic rays to be observed on years, the intensity of the cosmer $σ = 2 \times 10^{- 3} erg / {cm}^{2} \sec$ . the earth should be of the order $σ = 2 \times 3 \times 10^{- 3} erg / {cm}^{2}$ The observational values are With all reserve we advance the sec . (Millikan, Regener). With the transitions from view that supernovae represent which in their final stages ordinary stars into neursely packed neutrons. nova therefore emits during if supernovae initially are consist of extremely closely packed neutrons $E_{τ} \geq 10^{5} L_{τ} = 3.78 \times 10^{53} ergs$ . If supernovae initially are

On Massive Neutron Cores

J. R. Oppenheimer and G. M. Volkoff

Department of Physics, University of California, Berkeley, California
(Received January 3, 1939)
It has been suggested that, when the pressure within stellar matter becomes high enough, a new phase consisting of neutrons will be formed. In this paper we study the gravitational equilibrium of masses of neutrons, using the equation of state for a cold Fermi gas, and general relativity. For masses under

\frac{1}{3} ⊙

only one equilibrium solution exists, which is approximately described by the nonrelativistic Fermi equation of state and Newtonian gravitational theory. For masses

\frac{1}{3} ⊙ < m < \frac{3}{4} ⊙

two solutions exist, one stable and quasi-Newtonian, one more condensed, and unstable. For masses greater than

\frac{3}{4} ⊙

there are no static equilibrium solutions. These results are qualitatively confirmed by comparison with suitably chosen special cases of the analytic solutions recently discovered by Tolman. A discussion of the probable effect of deviations from the Fermi equation of state suggests that actual stellar matter after the exhaustion of thermonuclear sources of energy will, if massive enough, contract indefinitely, although more and more slowly, never reaching true equilibrium.

Figure 24.1.
Two important arrivals on the astrophysical scene: the neutron star (1933) and the black hole

(1795, 1939)

.
No proper account of either can forego general relativity.

DU MONDE,

\section*{EXPOSition
\section*{EXPOSition
,}

Par Pierre-Simon Laplace, de l'Institut National de France, et du Bureau des Longitudes.

TOME SECOND.

A PARIS,
De l'Imprimerie du Cercle-Social, rue du
Théàtre Français,

N^{\circ}

. 4 .
s'an IV de la Rétubliquin Finggitse.

(305)

aussi sensibles à la distance qui nous etr separe ; ct combien ils doivent surpasser ceux que nous observons à la surface du soleil ? Tous ces corps devenus invisibles, sont à la même place où ils ont été observés, puisquiis n'en ont point changé, durant leur apparition; il existe donc dans les espaces célestes, des corps obscurs aussi considérables, et peut être en aussi grand nombre, que les. étoiles. Un astre lumineux de même densité que la terre, et dont le diamètre serait deux cents cinquante fois plus grand que celui du soleil, ne laisserait en vertu de son attraction, parvenir aucun de ses rayons jusqu"à nous; il est donc possible que les plus grands corps lumineux de l'univers, soient par cela même, invisibles. Une etoile qui, sans être de cette grandeur, surpasserait considerablement le soleil ; affaiblirait sensiblement la vitesse de la lumière, et augmenterait ainsi l'étendue de son aberration. Cette différence dans l'aberration des étoiles ; un catalogue de celles qui ne font que paraitre, et leur position observee au moment de leur éclat passager; la dè e termination de toutes les étoiles changeantes,

Tome IL v

On Continued Gravitational Contraction

J. R. Oppenheimer and H. SnyderUniversity of California, Berkeley, California

(Received July 10, 1939)
When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely. In the present paper we study the solutions of the gravitational field equations which describe this process. In I, general and qualitative arguments are given on the behavior of the metrical tensor as the contraction progresses: the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles. In II, an analytic solution of the field equations confirming these general arguments is obtained for the case that the pressure within the star can be neglected. The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius.
"Final state of stellar evolution," and "cold, catalyzed matter" defined

Equation of state for cold, catalyzed matter

What does one mean in principle by the term "the final state of stellar evolution"? Start with a star containing a given number,

A

, of baryons and let it evolve to the absolute, burned-out end point of thermonuclear combustion (minimum massenergy possible for the

A

-baryon system). If the normal course of thermonuclear combustion is too slow, speed it up by catalysis. If an explosion occurs, collect the outgoing matter, extract its kinetic energy, and let it fall back onto the system. Repeat this operation as many times as needed to arrive at burnout (cold

{Fe}^{56}

for the part of the system under modest pressure; other nuclear species in the region closer to the center; "cold matter catalyzed to the end point of thermonuclear combustion" throughout). End up finally with the system in its absolutely lowest energy state, with all angular momentum removed and all heat extracted, so that it sits at the absolute zero of temperature and has zero angular velocity. Such a "dead" system, depending upon its mass and prior history (two distinct energy minima for certain

A

-values), ends up as a cold stellar configuration (neutron star, or "white" dwarf), or as a "dead" black hole.

The analysis of a cold stellar configuration demands an equation of state. The temperature is fixed at zero; the nuclear composition in principle is specified uniquely by the density; and therefore the pressure is also fixed uniquely once the density has been specified [equation of state

p (ρ)

for "cold catalyzed matter"].

The white dwarfs and neutron stars observed by astronomers are not really built of cold catalyzed matter. However, the matter in them is sufficiently near the end point of thermonuclear evolution and sufficiently cold that it can be idealized with fair accuracy as cold and catalyzed (see §23.4).
The equation of state,

ρ (p)

, for cold catalyzed matter is shown graphically in Figure 24.2. This version of the equation of state was constructed by Harrison and Wheeler in 1958. Other versions constructed more recently [see Cameron (1970) and Baym, Bethe, and Pethick (1971) for references] are almost identical to the Harrison-Wheeler version at densities well below nuclear densities,

ρ < 3 \times 10^{13}

g / {cm}^{3}

. At nuclear and supernuclear densities, all versions differ because of differing assumptions about nucleon-nucleon interactions. Along with the equation of state, in Figure 24.2 are shown properties of the models of cold stars constructed from this equation of state by integrating numerically the equations of structure (23.28).
The equation of state can be understood by following the transformations that occur as a sample of cold catalyzed matter is compressed to higher and higher densities. At each stage in the compression, each possible thermonuclear reaction is to be catalyzed to its endpoint and the resultant thermal energy is to be removed.
When the sample is at zero pressure, it is a ball of pure, cold

{Fe}^{56}

, since

{Fe}^{56}

is the most tightly bound of all nuclei. It has the density

7.86 g / {cm}^{3}

. As the sample is compressed, its internal pressure is provided at first by normal solid-state forces; but the atoms are soon squeezed so closely together that the electrons become quite oblivious of their nuclei, and begin to form a degenerate Fermi gas. By the time a density of

ρ = 10^{5} g / {cm}^{3}

has been reached, valence forces are completely negligible, the degenerate electron pressure dominates, and the compressibility index,

γ

(see legend for Figure 24.2), is

5 / 3

, the value for a nonrelativistically degenerate Fermi gas. Between

10^{5}

and

10^{7} g / {cm}^{3}

, the pressure-providing electrons gradually

Equation of state

Stellar models

Figure 24.2.
The Harrison-Wheeler equation of state for cold matter at the absolute end point of thermonuclear evolution, and the corresponding Harrison-Wakano-Wheeler stellar models. The equation of state is exhibited in the form of a plot of "compressibility index,"

γ = \frac{ρ + p}{p} \frac{d p}{d ρ}

as a function of density of mass-energy,

ρ

. (Small

γ

corresponds to easy compressibility.) The curve is parameterized by the logarithm of the pressure,

\log_{10} p

, in units of

g / {cm}^{3}

[same units as

ρ

; note that

p (g / {cm}^{3}) = (1 / c^{2}) \times p (dyne / {cm}^{2})]

. The chemical composition of the matter as a function of density is indicated as follows:

Fe, {Fe}^{56}

nuclei; A , nuclei more neutron rich than

{Fe}^{56}

; e, electrons; n, free neutrons; p, free protons.

The first law of thermodynamics [equation (22.6)], when applied to cold matter (zero entropy) says

d ρ / (ρ + p) = d n / n

; i.e.,

n = \frac{ρ + p}{μ_{Fe} / 56} \exp (- \int_{0}^{p} \frac{d p}{ρ + p})

Here

μ_{Fe}

, the rest mass of an

{Fe}^{56}

atom, is the ratio between

ρ + p \approx ρ

and

n / 56

in the limit of zero density. From this equation and a knowledge of

ρ (p)

-(see Figure)-one can calculate

n (p)

The equilibrium configurations are represented by curves of total mass-energy,

M

, versus radius,

R

. (

R

is defined such that

4 π R^{2}

is the star's surface area.) The

M (R)

curve is parameterized by the logarithm of the central density,

\log_{10} ρ_{c}

, measured in

g / {cm}^{3}

. Only configurations along two branches of the curve are stable against small perturbations and can therefore exist in nature: the white dwarfs, with

\log_{10} ρ_{c} < 8.38

, and the neutron stars, with

13.43 < \log_{10} ρ_{c} < 15.78

(see Box 26.1).

For greater detail on both the equation of state and the equilibrium configurations, see Harrison, Thorne, Wakano, and Wheeler (1965); also, for an updated table of the equation of state, see Hartle and Thorne (1968).
become relativistically degenerate, and

γ

approaches

4 / 3

. Above

ρ = 1.4 \times 10^{7}

g / {cm}^{3}

, the rest mass of

62 {Fe}_{26}^{56}

nuclei, plus the rest mass of 44 electrons, plus the rather large Fermi kinetic energy of 44 electrons at the top of the Fermi sea, exceeds the rest mass of

56 {Ni}_{28}^{62}

nuclei. Consequently, as the catalyzed sample of matter is compressed past

ρ = 1.4 \times 10^{7} g / {cm}^{3}

, the nuclear reaction

\begin{aligned} (24.1) & 62 {Fe}_{26}^{56} (\binom{highly compressed}{neutral atoms}) ⟶ \\ 56 {Ni}_{28}^{62} (\binom{highly compressed}{neutral atoms}) \end{aligned}

goes to its end point, with a release of energy. As the compression continues beyond this point, the rising Fermi energy of the electrons induces new nuclear reactions similar to (24.1), but involving different nuclei. In these reactions more and more electrons are swallowed up to form new nuclei, which are more and more neutronrich. When the density reaches

ρ = 3 \times 10^{11} g / {cm}^{3}

, the nuclei are so highly neutronrich

(Y_{39}^{122})

that neutrons begin to drip off them. The matter now becomes highly compressible for a short time

(3 \times 10^{11} ≦ ρ ≦ 4 \times 10^{11})

, since most of the remaining electrons are swallowed up very rapidly by the dripping nuclei. Above

ρ \sim 4 \times 10^{11} g / {cm}^{3}

free neutrons become plentiful and their degeneracy pressure exceeds that of the electrons. Further compression to

ρ \sim 10^{13} g / {cm}^{3}

completely disintegrates the remaining nuclei, leaving the sample almost pure neutrons with

γ = 5 / 3

, the value for a nonrelativistically degenerate Fermi gas. Intermixed with the neutrons are just enough degenerate electrons to prevent the neutrons from decaying, and just enough protons to maintain charge neutrality. Compression beyond

ρ \sim 10^{13} g / {cm}^{3}

pushes the sample into the domain of nuclear densities where the physics of matter is only poorly understood. This Harrison-Wheeler version of the equation of state ignores all nucleon-nucleon interactions at abd above nuclear densities; it idealizes matter as a noninteracting mixture of neutrons, protons, and electrons with neutrons dominating; and it shows a compressibility index of

5 / 3

while the neutrons are nonrelativistic, but

4 / 3

after they attain relativistic Fermi energies. Other versions of the equation of state attempt to take into account the nucleonnucleon interactions in a variety of ways [see Cameron (1970), Baym, Bethe, and Pethick (1971), and many references cited therein].

Corresponding to each value of the central density,

ρ_{c}

, there is one stellar equilibrium configuration. Equilibrium, yes; but is the equilibrium stable? Stability studies (Chapter 26, especially Box 26.1 ) show that many of the models are unstable against small radial perturbations, which lead to gravitational collapse. Only white-dwarf stars in the range

\log_{10} ρ_{c} < 8.4

and neutron stars in the range

13.4 ≦ \log_{10} ρ_{c} ≦ 15.8

are stable. Instability for the region of

\log_{10} ρ_{c}

values between 8.4 and 13.4 is caused by a combination of (1) relativistic strengthening of the gravitational forces, and (2) high compressibility of the matter due to electron capture and neutron drip by

Equilibrium configurations for cold, catalyzed matter:
(1) forms and stability
the atomic nuclei. Neutron stars are stable for a simple reason. Neutron-dominated matter is so difficult to compress that even the relativistically strengthened gravitational forces cannot overcome it. Above

\log_{10} ρ_{c} \sim 15.8

, the gravitational forces become strong enough to win out over the pressure of the nuclear matter, and the stars ate all unstable. [See Gerlach (1968) for the possibility-which, however, he rates as unlikely-that there might exist a third family of stable equilibrium configurations, additional to white dwarfs and neutron stars.]

The white-dwarf stars have masses below

1.2 M_{⊙}

and radii between

\sim 3000

and

\sim 20, 000 km

. They are supported almost entirely by the pressure of the degenerate electron gas. Relativistic deviations from Newtonian structure are only a fraction of a per cent, but relativistic effects on stability and pulsations are important from

ρ_{c} \approx 10^{8} g / {cm}^{3}

to the upper limit of the white-dwarf family at

ρ_{c} = 10^{8.4} g / {cm}^{3} [see

, e.g., Faulkner and Gribbin (1968)]. The properties of white-dwarf models are fairly independent of whose version of the equation of state is used in the calculations.

The properties of neutron stars are moderately dependent on the equation of state used. However, all versions lead to upper and lower limits on the mass and central density. The correct lower limits probably lie in the range

\begin{aligned} (24.2) & 13.4 ≦ \log_{10} ρ_{c min} \leq 14.0 \\ 0.05 M_{⊙} \leq M_{min} \leq 0.2 M_{⊙} \end{aligned}

the correct upper limits are probably in the range

\begin{aligned} (24.3) & 15.0 ≲ \log_{10} ρ_{c max} ≲ 16.0, \\ 0.5 M_{⊙} ≦ M_{max} ≲ 3 M_{⊙} \end{aligned}

[see Rhoades (1971)]. Neutron stars typically have radii between

\sim 6 km

and

\sim 100

km . Relativistic deviations from Newtonian structure are great, sometimes more than 50 per cent.

It appears certain that no cold stellar configuration can have a mass exceeding

\sim 5 M_{⊙} [

Rhoades (1971)] (1.2

M_{⊙}

according to the Harrison-Wheeler equation of state, Figure 24.2). Any star more massive than this must reduce its mass below this limit if it is to fade away into quiet obscurity, otherwise relativistic gravitational forces will eventually pull it into catastrophic gravitational collapse past white-dwarf radii, past neutron-star radii, and into a black hole a few kilometers in size (see Part VII).

§24.3. PULSARS

Theory predicts that, when a star more massive than the Chandrasekhar limit of

1.2 M_{⊙}

has exhausted the nuclear fuel in its core and has compressed its core to white-dwarf densities, an instability pushes the star into catastrophic collapse. The

Birth of a neutron star by stellar collapse

Dynamics of a newborn neutron star

Neutron star as a pulsar

Pulsar radiation as a tool for studying neutron stars
core implodes upon itself until nucleon-nucleon repulsion halts the implosion. The result is a neutron star, unless the core's mass is so great that gravity overcomes the nucleon-nucleon repulsion and pulls the star on in to form a black hole. Not all the star's mass should become part of the neutron star or black hole. Much of it, perhaps most, can be ejected into interstellar space by the violence that accompanies the collapse-violence due to flash nuclear burning, shock waves, and energy transport by neutrinos ("stick of dynamite in center of star, ignited by collapse").

The collapsed core holds more interest for gravitation theory than the ejected envelope. That core, granted a mass small enough to avoid the black-hole fate, will initially be a hot, wildly pulsating, rapidly rotating glob of nuclear matter with a strong, embedded magnetic field (see Figure 24.3). The pulsations must die out quickly. They emit a huge flux of gravitational radiation, and radiation reaction damps them in a characteristic time of

\sim 1

second [see Wheeler (1966); Thorne (1969a)]. Moreover, the pulsations push and pull elementary particle reactions back and forth by raising and lowering the Fermi energies in the core's interior; these particle reactions can convert pulsation energy into heat at about the same rate as the pulsation energy is radiated by gravity. [See Langer and Cameron (1969); also §11.5 of Zel'dovich and Novikov (1971) for details and references.]

The result, after a few seconds, is a rapidly rotating centrifugally flattened neutron star with a strong (perhaps

10^{12}

gauss) magnetic field; all the pulsations are gone. If the star is deformed from axial symmetry (e.g., by centrifugal forces or by a nonsymmetric magnetic field), its rotation produces a steady outgoing stream of gravitational waves, which act back on the star to remove rotational energy. Whether or not this occurs, the rotating magnetic field itself radiates electromagnetic waves. They slow the rotation and transport energy into the surrounding, exploding gas cloud (nebula). [See Pacini (1968), Goldreich and Julian (1968), and Ostriker and Gunn (1969) for basic considerations.]
Somehow, but nobody understands in detail how, the rotating neutron star beams coherent radio waves and light out into space. Each time the beam sweeps past the Earth optical and radio telescopes see a pulse of radiation. The light is emitted synchronously with the radio waves, but the light pulses reach Earth earlier (

\sim 1

second for the pulsar in the crab nebula) because of the retardation of the radio waves by the plasma along the way. This is the essence of the 1973 theory of pulsars, accepted by most astrophysicists.

Although the mechanism of coherent emission is not understood, the pulsar radiation can nevertheless be a powerful tool in the experimental study of neutron stars. Anything that affects the stellar rotation rate, even minutely (fractional changes as small as

10^{- 9}

) will produce measurable irregularities in the timing of the pulses at Earth. If the star's crust and mantle are crystalline, as 1973 theory predicts, they may be subject to cracking, faulting, or slippage ("starquake") that changes the moment of inertia, and thence the rotation rate. Debris falling into the star will also change its rotation. Whichever the cause, after such a disturbance the star may rotate differentially for awhile; and how it returns to rigid rotation may depend on such phenomena as superfluidity in its deep interior. Thus, pulsar-timing data may eventually give information about the interior and crust of the neutron star, and

Figure 24.3.
"Collapse, pursuit, and plunge scenario" [schematic from Ruffini and Wheeler (1971b)].

A star with white-dwarf core (A), slowly rotating,
evolves by straightforward astrophysics,
arrives at the point of gravitational instability,
collapses, and
ends up as a rapidly spinning neutron-star pancake ( $B, B^{'}$ ).
It then fragments (C) because it has too much angular momentum to collapse into a single stable object. If the substance of the neutron-star pancake were an incompressible fluid, the fragmentation would have a close tie to well-known and often observed phenomena ("drop formation"). However, the more massive a neutron star is, the smaller it is, so one's insight into this and subsequent stages of the scenario are of necessity subject to correction or amendment. One can not today guarantee that fragmentation takes place at all; nevertheless, fragmentation will be assumed in what follows.
The fragments dissipate energy and angular momentum via gravitational radiation.
One by one as they revolve they coalesce ("pursuit and plunge scenario").
In each such plunge a pulse of gravitational radiation emerges.
Fragments of debris fall onto the coalesced objects (neutron stars or black holes, as the case may be), changing their angular momenta.
Eventually the distinct neutron stars or black holes or both unite into one such collapsed object with a final pulse of gravitational radiation.
The details of the complete scenario differ completely from one evolving star to another, depending on
the mass of its core, and
the angular momentum of this core.
An entirely different kind of picture therefore has to be drawn for altered values of these two parameters.
Even for the values of these parameters adopted in the drawing, the present picture can at best possess only qualitative validity.
Detailed computer analysis would seem essential for any firm prediction about the course of any selected scenario.
thence (by combination with theory) about its mass and radius. These issues are discussed in detail in a review article by Ruderman (1972) as well as in Zel'dovich and Novikov (1971).

§24.4. SUPERMASSIVE STARS AND STELLAR INSTABILITIES

Theory of the stability of Newtonian stars

When a Newtonian star of mass

M

oscillates adiabatically in its fundamental mode, the change in its radius,

δ R

, obeys a harmonic-oscillator equation,

\begin{matrix} (24.4) & M δ \ddot{R} = - k δ R, \end{matrix}

with a "spring constant"

k

that depends on the star's mean adiabatic index

{\bar{Γ}}_{1}

[recall:

Γ_{1} \equiv (n / p) (\partial p / \partial n)_{const. entropy}

], on its gravitational potential energy

Ω

, on the trace

I = \int ρ r^{2} d V

of the second moment of its mass distribution, and on its mass

M

\begin{matrix} (24.5) & k = 3 M ({\bar{Γ}}_{1} - 4 / 3) | Ω | / I \end{matrix}

(See Box 24.2). If

{\bar{Γ}}_{1} > 4 / 3

the Newtonian star is stable and oscillates; if

{\bar{Γ}}_{1} < 4 / 3

the star is unstable and either collapses or explodes, depending on its initial conditions and overall energetics. This result is a famous theorem in Newtonian stellar theory-but it is relevant only for adiabatic oscillations.

Box 24.2 OSCILLATION OF A NEWTONIAN STAR

The following is a volume-averaged analysis of the lowest mode of radial oscillation. Such analyses are useful in understanding the qualitative behavior and stability of a star. [See Zel'dovich and Novikov (1971) for an extensive exploitation of them.] However, for precise quantitative results, one must perform a more detailed analysis [see, e.g., Ledoux and Walraven (1958); also Chapter 26 of this book].

Let $M =$ star's total mass
$R =$ star's radius
$\bar{ρ} =$ mean density $= (3 / 4 π) M / R^{3}$
$\bar{p} =$ mean pressure
${\bar{Γ}}_{1} =$ mean adiabatic index $= (\bar{n} / \bar{p}) (\partial \bar{p} / \partial \bar{n})_{adiabatic}$
$= (\bar{ρ} / \bar{p}) (\partial \bar{p} / \partial \bar{ρ})_{adiabatic}$ in Newtonian limit, where $ρ =$ const. $\times n$ .
Then the mean pressure-buoyancy force ${\bar{F}}_{buoy}$ and the counterbalancing gravitational force ${\bar{F}}_{grav}$ in the equilibrium star are

\begin{aligned} {\bar{F}}_{buoy} & = \bar{p} / R \\ = {\bar{F}}_{grav} = \bar{ρ} M / R^{2} = (4 π / 3) {\bar{ρ}}^{2} R . \end{aligned}

When the oscillating star has expanded or contracted so its radius is $R + δ R$ , then its mean density will have changed to

\bar{ρ} + δ \bar{ρ} = (3 / 4 π) M [R^{- 3} + δ (R^{- 3})] = \bar{ρ} - 3 (\bar{ρ} / R) δ R,

and its mean pressure will be

\bar{p} + δ \bar{p} = \bar{p} + (\bar{p} / \bar{ρ}) {\bar{Γ}}_{1} δ \bar{ρ} = \bar{p} - 3 ({\bar{Γ}}_{1} \bar{p} / R) δ R .

The corresponding changes in the forces will be

\begin{aligned} δ {\bar{F}}_{buoy} = \frac{δ \bar{p}}{R} - \frac{\bar{p}}{R^{2}} δ R = - (3 Γ_{1} + 1) \frac{\bar{p}}{R} \frac{δ R}{R} = - (3 {\bar{Γ}}_{1} + 1) {\bar{F}}_{buoy} (\frac{δ R}{R}) \\ δ {\bar{F}}_{grav} = (\frac{4 π}{3}) (2 \bar{ρ} R δ \bar{ρ} + {\bar{ρ}}^{2} δ R) = (\frac{4 π}{3} {\bar{ρ}}^{2} R) (- 5 \frac{δ R}{R}) = - 5 {\bar{F}}_{grav} (\frac{δ R}{R}) \end{aligned}

Consequently, the restoring force will be (recall:

{\bar{F}}_{buoy} = {\bar{F}}_{grav}

)

δ {\bar{F}}_{grav} - δ {\bar{F}}_{buoy} = 3 ({\bar{Γ}}_{1} - \frac{4}{3}) {\bar{F}}_{grav} \frac{δ R}{R} .

This restoring force produces an acceleration,

δ {\bar{F}}_{grav} - δ {\bar{F}}_{buoy} = - \bar{ρ} δ \ddot{R}

Hence, the equation of motion for the oscillations is

δ \ddot{R} = - 3 ({\bar{Γ}}_{1} - 4 / 3) (4 π / 3) \bar{ρ} δ R,

corresponding to a "spring constant"

k

and angular frequency of oscillation

ω

, given by

ω^{2} = 4 π ({\bar{Γ}}_{1} - 4 / 3) \bar{ρ}

, and

k = M ω^{2}

.
5. A more nearly exact analysis (see exercise 39.7 for details, or Box 26.2 for an alternative derivation) yields the improved formula

\begin{matrix} ω^{2} = 3 ({\bar{Γ}}_{1} - 4 / 3) | Ω | / I, \\ Ω = (\binom{star's self-gravitational}{energy}) = \frac{1}{2} \int ρ Φ d V = - \frac{1}{2} \int \frac{ρ ρ^{'}}{| x - x^{'} |} d V d V^{'}, \\ I = (\binom{trace of second moment of}{star's mass distribution}) = \int ρ r^{2} d V, \end{matrix}

for the square of the oscillation frequency.
6. Note that

{\bar{Γ}}_{1} > 4 / 3

corresponds to stable oscillations;

{\bar{Γ}}_{1} < 4 / 3

corresponds to exponentially developing collapse or explosion.

Stability theory predicts
"engine-driven oscillations" and quick death for stars of

M > 60 M_{⊙}

Possible existence of supermassive stars

Relativistic instabilities in a supermassive star

In a real star no oscillation is precisely adiabatic. The oscillations in temperature cause corresponding oscillations in the stellar opacity and in nuclear burning rates. These insert energy into or extract energy from the gas vibrations.

All main-sequence stars thus far observed and studied have masses below

60 M_{⊙}

. For such small masses, theory predicts low enough temperatures that gas pressure dominates over radiation pressure, and the adiabatic index is nearly that of nonrelativistic gas,

{\bar{Γ}}_{1} \approx 5 / 3

. Such stars vibrate stably. The net effect of the oscillating opacity and burning rate is usually to extract energy from the vibrations. Thus, they damp. (The vibrations of Cepheid variable stars are a notable exception.)
No one has yet seen a main-sequence star with mass above about

60 M_{⊙}

. This is explained as follows. For masses above

60 M_{⊙}

, the temperature should be so high that radiation pressure dominates over gas pressure, and the adiabatic index

{\bar{Γ}}_{1}

is only slightly above the value

4 / 3

for pure radiation. Consequently the "spring constant" of the star, although positive, is very small. On the inward stroke of an oscillation, the central temperature rises, and nuclear burning speeds up. (The nuclear burning rate goes as a very high power of the central temperature; for example, in a massive star HCNO burning releases energy at a rate

ε_{HCNO} \propto T_{c}^{11}

.) Because the spring constant is so small, the inward stroke lasts for a long time, and the enhanced nuclear burning produces a significant excess of thermal energy and pressure. Hence, on the outward stroke the star expands more vigorously than it contracted ("engine"). Successive vibrations are driven to higher and higher amplitudes. Eventually, calculations suggest, the star either explodes, or it ejects enough mass by its vigorous vibrations to drop below the critical limit of

M \sim 60 M_{⊙}

. Hence, stars of mass above

60 M_{⊙}

should not live long enough that astronomers could have a reasonable probability of discovering them.
Of course, this "engine action" does not prevent massive stars from forming, living a short time, and then disrupting themselves. Such a possibility is particularly intriguing for supermassive stars [

M

between

10^{3} M_{⊙}

and

10^{9} M_{⊙} \sim 0.01 \times

(mass of a galaxy)]. Although such stars may be exceedingly rare, by their huge masses and huge release of explosive energy they might play an important role in the universe. Moreover, it is conceivable that the oscillations of such stars, like those of Cepheid variables, might be sustained at large amplitudes for long times (a million years?), with nonlinear damping processes preventing their further growth.

Theory predicts that general relativistic effects should strongly influence the oscillations of a supermassive star. The increase in "gravitational force,"

δ F_{grav}

, acting on a shell of matter on the inward stroke is greater in general relativity than in Newtonian theory, and the decrease on the outward stroke is also greater. Consequently the "effective index"

Γ_{1 crit}

of gravitational forces is increased above the Newtonian value of

4 / 3

; thus,

(\begin{array}{l} fractional increase in \\ "pressure-like force of \\ gravity" per unit fractional \\ change in baryon-number \\ density \end{array}) \equiv Γ_{1 crit} = (4 / 3) + α (M / R) + O (M^{2} / R^{2})

,
where

α

is a constant of the order of unity that depends on the structure of the star (see Box 26.2). To resist gravity, one has only the elasticity of the relativistic material of the star:

\begin{matrix} (24.7) & (\begin{array}{l} fractional increase in \\ "pressure-like resisting \\ force" per unit fractional \\ change in baryon number \\ density \end{array}) = {\bar{Γ}}_{1} = {⟨ \frac{p}{n} {(\frac{\partial p}{\partial n})}_{s} ⟩}_{\begin{matrix} effective average \\ over star \end{matrix}} \end{matrix}

The effective spring constant for the vibrations of the star is governed by the delicate margin between these two indices:

\begin{aligned} k = (\binom{effective}{spring constant}) & = (\begin{matrix} contribution \\ of "elastic \\ forces" \end{matrix}) - (\binom{contribution}{of gravity}) \\ (24.8) & = 3 M ({\bar{Γ}}_{1} - Γ_{1 crit}) \frac{| Ω |}{I} \end{aligned}

(derivation in Chapter 26). The relativistic rise in the effective index of gravity above

4 / 3

[equation (24.6)] brings on the transition from stability (positive

k

; vibration) to instability (negative

k

; explosion or collapse) under conditions when one otherwise would have expected stability. For supermassive stars, Fowler and Hoyle (1964) show that

{\bar{Γ}}_{1} = 4 / 3 + ζ {(M / M_{⊙})}^{- 1 / 2}

where

ζ

is a constant of order unity. As a newly formed supermassive star contracts inward, heating up, but not yet hot enough to ignite its nuclear fuel, it approaches nearer and nearer to instability against collapse. Unless burning halts the contraction, collapse sets in at a radius

R_{crit}

given by

{\bar{Γ}}_{1} = 4 / 3 + ζ {(M / M_{⊙})}^{- 1 / 2} = Γ_{1 crit} = 4 / 3 + α M / R;

i.e.,

\begin{aligned} R & = (α / 2 ζ) {(M / M_{⊙})}^{1 / 2} \times (Schwarzschild Radius) \\ \sim 10^{4} \times (Schwarzschild Radius) if M = 10^{8} M_{⊙} \end{aligned}

The relativistic instability occurs far outside the Schwarzschild radius when the star is very massive. Relativity hardly modifies the star's structure at all; but because of the delicate balance between

δ {\bar{F}}_{grav}

and

δ {\bar{F}}_{buoy}

in the Newtonian oscillations (Box 24.2), tiny relativistic corrections to these forces can completely change the stability.

In practice, the story of a supermassive star is far more complicated than has been indicated here. Rotation can stabilize it against relativistic collapse for a while. However, after the star has lost all angular momentum in excess of the critical value

Temporary stabilization by rotation

Possible scenarios for evolution and death of a supermassive star

J_{crit} = M^{2}

("extreme Kerr limit"; see Chapter 33), and after it has contracted to near the Schwarzschild radius, rotation is helpless to stave off implosion. Depending on its mass and angular momentum, the star may ignite its fuel before or after relativistic collapse begins, and before or after implosion through the Schwarzschild radius. When the fuel is ignited, it can wreak havoc, because even if the star is not then imploding, its adiabatic index will be very near the critical one, and the burning may drive oscillations to higher and higher amplitudes. These processes are so complex that in 1973 one is far from having satisfactory analyses of them, but for reviews of what is known and has been done, the reader can consult Fowler (1966), Thorne (1967), and Zel'dovich and Novikov (1971).

The theory of stellar pulsations in general relativity is presented for Track-2 readers in Chapter 26 of this book.

§24.5. QUASARS AND EXPLOSIONS IN GALACTIC NUCLEI

Supermassive stars were first conceived by Hoyle and Fowler (1963a,b) as an explanation for explosions in the nuclei of galaxies. Shortly thereafter, when quasars were discovered, Hoyle and Fowler quite naturally appealed to their supermassive stars for an explanation of these puzzles as well. Whether galactic explosions or quasars are driven by supermassive stars remains a subject of debate in astronomical circles even as this book is being finished, in 1973. Hence, this book will avoid the issue except for the following remark.
Whatever is responsible for quasars and galactic explosions must be a machine of great mass (

M \sim 10^{6}

10^{10} M_{⊙}

) and small radius (light-travel time across the machine, as deduced from light variations, is sometimes less than a day). The machine might be a coherent object, i.e., a supermassive star; or it might be a dense mixture of ordinary stars and much gas. Actually these two possibilities may not be distinct. Star-star collisions in a dense cluster can lead to stellar coalescence and the gradual building up of one or more supermassive stars [Sanders (1970); Spitzer (1971); Colgate (1967)]. Thus, at one stage in its life, a galactic nucleus or quasar might be driven by collisions in a dense star cluster; and at a later stage it might be driven by a supermassive star; and at a still later stage that star might collapse to leave behind a massive black hole

(10^{6} - 10^{9} M_{⊙})

, but a black hole that is still "live" and active (Chapter 33).

§24.6. RELATIVISTIC STAR CLUSTERS

The normal astrophysical evolution of a galactic nucleus is estimated [Sanders (1970); Spitzer (1971)] to lead under some circumstances to a star cluster so dense that general relativity influences its structure and evolution. The theory of relativistic star clusters is closely related to that of relativistic stars, as developed in Chapter 23. A star is a swarm of gas molecules that collide frequently; a star cluster is a swarm of stars that collide rarely. But the frequency of collisions is relatively unim-
portant in a steady state. For the theory of relativistic star clusters, see:

§ 25.7

of this book; Zel'dovich and Podurets (1965); Fackerell, Ipser, and Thorne (1969); Chapter 12 of Zel'dovich and Novikov (1971); and references cited there. A relativistic star cluster is a latent volcano. No future is evident for it except to evolve with enormous energy release to a massive black hole, either by direct collapse (possibly a star at a time) or by first coalescing into a supermassive star that later collapses.

снартев 25

THE "'PIT IN THE POTENTIAL" AS THE CENTRAL NEW FEATURE OF MOTION IN SCHWARZSCHILD GEOMETRY

"Eccentric, intervolved, yet regularThen most, when most irregular they seem; And in their motions harmony divine"

This chapter is entirely Track 2, except for Figures 25.2 and 25.6, and Boxes 25.6 and 25.7 (pp. 639, 660, 674, and 677), which Track-1 readers should peruse for insight and flavor. No earlier Track-2 material is needed as preparation for it.
§25.2 (symmetries) is needed as preparation for Box 30.2 (Mixmaster cosmology). The rest of the chapter is not essential for any later chapter, but it will be helpful in understanding
(1) Chapters 31-34
(gravitational collapse and black holes), and
(2) Chapter 40 (solar-system experiments).

§25.1. FROM KEPLER'S LAWS TO THE EFFECTIVE POTENTIAL FOR MOTION IN SCHWARZSCHILD GEOMETRY

No greater glory crowns Newton's theory of gravitation than the account it gives of the principal features of the solar system: a planet in its motion sweeps out equal areas in equal times; its orbit is an ellipse, with one focus at the sun; and the cube of the semimajor axis,

a

, of the ellipse, multiplied by the square of the average angular velocity of the planet in its orbit (

ω = 2 π /

period) gives a number with the dimensions of a length, the same number for all the planets (Box 25.1), equal to the mass of the sun:

M = ω^{2} a^{3}

Exactly the same is true for the satellites of Jupiter (Figure 25.1), and of the Earth (Box 25.1 ), and true throughout the heavens. What more can one possibly expect of Einstein's theory of gravity when it in its turn grapples with this centuries-old theme of a test object moving under the influence of a spherically symmetric center of attraction? The principal new result can be stated in a single sentence: The particle is governed by an "effective potential" (Figure 25.2 and

§ § 25.5, 25.6

) that possesses not only (1) the long distance

- M / r

attractive behavior and (2) the shorter distance
(angular momentum)

)^{2} / r^{2}

repulsive behavior of Newtonian gravitational theory, but also (3) at still shorter distances a pit in the potential, which (1) captures a particle that comes too close; (2) establishes a critical distance of closest approach for this black-hole capture process; (3) for a particle that approaches this critical point without crossing it, lengthens the turn-around time as compared to Newtonian expectations; and thereby (4) makes the period for a radial excursion longer than the period of a revolution; (5) causes an otherwise Keplerian orbit to precess; and (6) deflects a fast particle and a photon through larger angles than Newtonian theory would predict.

The pit in the potential being thus the central new feature of motion in Schwarzschild geometry and the source of major predictions (Box 25.2), it is appropriate to look for the most direct road into the concept of effective potential and its meaning

and application. In this search no guide is closer to hand than Newtonian mechanics.
Analytic mechanics offers several ways to deal with the problem of motion in a central field of force, and among them are two of central relevance here: (1) the world-line method, which includes second-order differential equations of motion, Lagrange's equations, search for constants of integration, reduction to first-order equations, and further integration in rather different ways according as one wants the shape of the orbit,

θ = θ (r)

, or the time to get to a given point on the world line,

t = t (r)

; and (2) the wave-crest method, otherwise known as the "eikonal method" or "Hamilton-Jacobi method," which gives the motion by the condition of "constructive interference of wave crests," thus making a single leap from the Hamilton-Jacobi equation to the motion of the test object. Both methods are em-

Figure 25.1.
Jupiter's satellites, as followed from night to night with field glasses or telescope, provide an opportunity to check for oneself the central ideas of gravitation physics in the Newtonian approximation (distances large compared to Schwarzschild radius). For the practically circular orbits of these satellites, Kepler's law becomes

M^{1} = ω^{2} r^{3}

("1-2-3 principle") and the velocity in orbit is

β = ω r

. Out of observations on any two of the quantities

β, M, ω, r

, one can find the other two. (In the opposite limiting case of two objects, each of mass

M

, going around their common center of gravity with separation

r

, one has

M = ω^{2} r^{3} / 2, β = ω r / 2

). The configurations of satellites I-IV of Jupiter as given here for December 1964 (days

0.0, 1.0, 2.0

, etc. in "universal time," for which see any good dictionary or encyclopedia) are taken from The American Ephemeris and Nautical Almanac for 1964 [U.S. Government Printing Office (1962)].

Box 25.1 MASS FROM MEAN ANGULAR FREQUENCY AND SEMIMAJOR AXIS: $M = ω^{2} a^{3}$

Appropriateness of Newtonian analysis shown by smallness of mass (or "halfSchwarzschild radius" or "extension of the pit in the potential") as listed in last column compared to the semimajor axis

a

in the next-to-last column. Basic data from compilation of Allen (1963).

Object	Period $^{a}$ (days)	$ω ({cm}^{- 1})$	$a (cm)$	$ω^{2} a^{3} (cm)$
Planets
Mercury	87.9686	$275.8 \times 10^{- 19}$	$0.5791 \times 10^{13}$	$1.477 \times 10^{5}$
Venus	224.700	107.95	1.0821	1.477
Earth	365.257	66.41	1.4960	1.477
Mars	686.980	35.31	2.2794	1.477
Jupiter	4332.587	5.599	7.783	1.478
Saturn	10759.20	2.255	14.27	1.477
Uranus	30685	0.7905	28.69	1.476
Neptune	60188	0.4030	44.98	1.478
Pluto	90700	$0.2674 \times 10^{- 19}$	$59.00 \times 10^{13}$	$1.469 \times 10^{5}$
Major satellites of Jupiter
Io	1.769138	$13.711 \times 10^{- 16}$	$0.422 \times 10^{11}$	141.3
Europa	3.551181	6.831	0.671	141.0
Ganymede	7.154553	$3.391 \sim 10^{- 16}$	1.070	140.8
Callisto	16.689018	$1.454 \times 10^{- 16}$	$1.883 \times 10^{11}$	141.1
Two satellites of Earth
OSO $5^{b}$	95.6 min .	$3.65 \times 10^{- 14}$	$6.916 \times 10^{8}$	0.442
Moon	27.32	$0.888 \times 10^{- 16}$	$3.84 \times 10^{10}$	0.446

^{a}

Sidereal period: time to make one revolution relative to fixed stars.

^{b}

Orbiting scientific observatory launched Jan. 22, 1969, to observe x-rays and ultraviolet radiation from the sun. Perigee 531 km , apogee 560 km , above earth.

SOME TYPICAL MASSES AND TIMES IN CONVENTIONAL AND GEOMETRIC UNITS. Conversion factor for mass,

G / c^{2} = 0.742 \times 10^{- 28} cm / g

Mass	Galaxy	Sun	Jupiter	Earth
$M_{conv} (g)$	$2.2 \times 10^{44}$	$1.989 \times 10^{33}$	$1.899 \times 10^{30}$	$5.977 \times 10^{27}$
$M (cm)$	$1.6 \times 10^{16}$	$1.47 \times 10^{5}$	112	0.444

Conversion factor for time,

c = 2.998 \times 10^{10} cm / \sec

. One sidereal year

= 365.256

days or

3.1558 \times 10^{7}

sec.

Period	1 sec	1 min	1 hr	1 day
$ω_{conv} (\sec^{- 1})$	6.28	$1.046 \times 10^{- 1}$	$1.75 \times 10^{- 3}$	$7.28 \times 10^{- 5}$
$ω ({cm}^{- 1})$	$2.09 \times 10^{- 10}$	$3.48 \times 10^{- 12}$	$5.80 \times 10^{- 14}$	$2.42 \times 10^{- 15}$
	1 week	1 month	1 year
	$1.04 \times 10^{- 5}$	$2.39 \times 10^{- 6}$	$1.99 \times 10^{- 7}$
	$3.46 \times 10^{- 16}$	$7.95 \times 10^{- 17}$	$6.63 \times 10^{- 18}$

Figure 25.2
Effective potential for motion of a test particle in the Schwarzschild geometry of a concentrated mass

M

. Energy, in units of the rest mass

μ

of the particle, is denoted

\tilde{E} = E / μ

; angular momentum,

\tilde{L} = L / μ

. The quantity

r

denotes the Schwarzschild

r

coordinate. The effective potential (also in units of

μ

) is defined by equation (25.16) or, equivalently, by the equation

{(\frac{d r}{d τ})}^{2} + {\tilde{V}}^{2} (r) = {\tilde{E}}^{2}

(see also

$ 25.5

) and has the value

\bar{V} = {[(1 - 2 M / r) (1 + {\bar{L}}^{2} / r^{2})]}^{1 / 2} .

It represents that value of

\tilde{E}

at which the radial kinetic energy of the particle, at

r

, reduces to zero (

\tilde{E}

-value that makes

r

into a "turning point":

\bar{V} (r) = \bar{E}

. Note that one could equally well regard

{\bar{V}}^{2} (r)

as the effective potential, and define a turning point by the condition

{\tilde{V}}^{2} = {\tilde{E}}^{2}

. Which definition one chooses depends on convenience, on the intended application, on the tie to the archetypal differential equation

\frac{1}{2} {\dot{x}}^{2} + V (x) = E

, and on the stress one wishes to put on correspondence with the effective potential of Newtonian theory). Stable circular orbits are possible (representative point sitting at minimum of effective potential) only for

\tilde{L}

values in excess of

2 \sqrt{3} M

. For any such fixed

L

value, the motion departs slightly from circularity as the energy is raised above the potential minimum (see the two heavy horizontal lines for

\tilde{L} = 3.75 M

). In classical physics, the motion is limited to the region of positive kinetic encrgy. In quantum physics, the particle can tunnel through the region where the kinetic energy, as calculated classically, is negative (dashed prolongations of heavy horizontal lines) and head for the "pit in the potential" (capture by black hole). Such tunneling is absolutely negligible when the center of attraction has any macroscopic dimension, but in principle becomes important for a black hole of mass

10^{17} g

(or

10^{- 11} cm

) if such an object can in principle exist.
The diagram at the right gives values of the minimum and maximum of the potential as they depend on the angular momentum of the test particle. The roots of

\partial \tilde{V} / \partial r

are given in terms of the "reduced angular momentum parameter"

L^{†} = \tilde{L} / M = L / M μ

\begin{matrix} r = \frac{6 M}{1 + {(1 - 12 / L^{† 2})}^{1 / 2}}, \\ {\tilde{E}}^{2} = \frac{(L^{+ 2} + 36) + (L^{† 2} - 12) {(1 - 12 / L^{† 2})}^{1 / 2}}{54} \\ [= 8 / 9 for L^{†} = (12)^{1 / 2}; 1 for L^{†} = 4; (L^{† 2} / 27) + (1 / 3) + (1 / L^{+ 2}) \\ + \dots for L^{†} ⟶ \infty] \end{matrix}

(plus root for maximum of the effective potential; minus root for minimum; see exercise 25.18 ).

Box 25.2 MOTION IN SCHWARZSCHILD GEOMETRY REGARDED AS A CENTRAL POINT OF DEPARTURE FOR MAJOR APPLICATIONS OF EINSTEIN'S GEOMETRODYNAMICS

Newtonian effect of sun on planets and of earth on moon and man.
Bending of light by sun.
Red shift of light from sun.
Precession of the perihelion of Mercury around the sun.
Capture of a test object by a black hole as simple exemplar of gravitational collapse.
Dynamics of Friedmann universe derived from model of Schwarzschild "lattice universe." Lattice universe is constructed from 120 or some other magic number of concentrations of mass, each mass in an otherwise empty lattice cell of its own. Each lattice cell, though actually polygonal, is idealized (see Wigner-Seitz approximation of solid-state physics) as spherical. A test object at the interface between two cells falls toward the center of each [standard radial motion in Schwarzschild geometry; see discussion following equation (25.27). Therefore the two masses fall toward each other at a calculable rate. From this simple argument follows the entire dynamics of the closed 3 -sphere lattice universe, in close concord with the predictions of the Friedmann model [see Lindquist and Wheeler (1957)].
Perturbations of Schwarzschild geometry, I. Gravitational waves are incident on, scattered by, and captured into a black hole. Waves with wavelength short compared to the Schwarzschild radius can be analyzed to good approximation by the methods of geometric optics (exercises 35.15 and 35.16), as employed in this chapter to treat the motions of particles and photons. For longer wavelengths, there occur important physical-optics corrections to this
geometric-optics idealization (see $§ 35.8$ and exercises 32.10, 32.11). Similar considerations apply to electromagnetic and de Broglie waves.
Lepton number for an electron in its lowest quantum state in the geometry ("gravitational field of force") of a black hole is calculated to be transcended (capture of the electron!) or not according as the mass of this black hole is large or small compared to a certain critical mass $M_{* e} = M^{* 2} / m_{e} (\sim 10^{17} g$ or $10^{- 11} cm)$ [Hartle (1971, 1972); Wheeler (1971b,c); Teitelboim (1972b,c)]. Similarly (with another value for the critical mass) for conservation of baryon number [Bekenstein (1972a,b), Teitelboim (1972a)]. To analyze "transcendence or not" one must solve quantum-mechanical wave equations, of which the Hamilton-Jacobi equation for particle and photon orbits is a classical limit. These quantum wave equations contain effective potentials identical-aside from spin-dependent and wavelength-dependent corrections-to the effective potentials for particle and photon motion.
Perturbations of Schwarzschild geometry, II. Those small changes in standard Schwarzschild black-hole geometry which remain stationary in time describe the alterations in a "dead" black hole that make it into a "live" black hole, one endowed with angular momentum as well as mass (see Chapter 33). To analyze such changes in black-hole geometry, one must again solve wave equations, but wave equations which are now classical. Once more the wave equations are closely related to the Hamilton-Jacobi equation, and their effective potentials are close kin to those for particle motion.
ployed here in turn because each gives special insights. The Hamilton-Jacobi method (Box 25.3) leads quickly to the major results of interest (Box 25.4), and it has a close tie to the quantum principle. The world-line method ( $$ $ 25.2, 25.3, 25.4$ ) starts with the geodesic equations of motion themselves. It provides a more familiar way into the subject for a reader not acquainted with the Hamilton-Jacobi approach. Moreover, in attempting to solve the geodesic equations of motion, one must analyze symmetry properties of the geometry, an enterprise that continues to pay dividends when one moves from Schwarzschild geometry to Kerr-Newman geometry (Chapter 33), and from Friedmann cosmology (Chapter 27) to more general cosmologies (Chapter 30).
(continued on page 650)

Box 25.3 THE HAMILTON-JACOBI DESCRIPTION OF MOTION: NATURAL BECAUSE RATIFIED BY THE QUANTUM PRINCIPLE

Purely classical (nonquantum).
Originated with William Rowan Hamilton out of conviction that mechanics is similar in its character to optics; that the "particle world line" of mechanics is an idealization analogous to the "light ray" of geometric optics. Localization of energy of light ray is approximate only. Its spread is governed by wavelength of light ("geometric optics"). Hamilton had glimmerings of same idea for particles: "quantum physics before quantum physics." The way that he and Jacobi developed to analyze motion through the Hamilton-Jacobi function $S (x, t)$ -to take the example of a dynamic system with only one degree of freedom, $x$ makes the leap from classical ideas to quantum ideas as short as one knows how to make it. Moreover, the real world is a quantum world. Classical mechanics is not born out of a vacuum. It is an idealization of and approximation to quantum mechanics.
Key idea is idealization to a particle wavelength so short that quantum-mechanical spread or uncertainty in location of particle (or spread of configuration coordinates of more complex system) is negligible. No better way was ever discovered to unite the spirit of quantum mechanics and the precision of location of classical mechanics.
Call Hamiltonian $H (p, x) = p^{2} / 2 m + V (x)$ . Call energy of particle $E$ . Then there is no way whatever consistent with the quantum principle to describe the motion of the particle in space and time. The uncertainty principle forbids (sharply defined energy $Δ E ⟶ 0$ , in $Δ E Δ t \geq ℏ / 2$ , implies uncertainty $Δ t ⟶ \infty$ ; also $Δ p ⟶ 0$ in $Δ p Δ x \geq ℏ / 2$ implies $Δ x ⟶ \infty)$ . The quantum-mechanical wave function is spread out over all space. This spread shows in the so-called semi-

Box 25.3 (continued)

classical or Wentzel-Kramers-Brillouin ["WKB"; see, for example, Kemble (1937)] approximation for the probability amplitude function,

\begin{matrix} (1) & ψ_{E} (x, t) = (\binom{slowly varying}{amplitude function}) e^{(i / h) S_{E} (x, t)} . \end{matrix}

5. It is of no help in localizing the probability distribution that

ℏ = 1.054 \times

10^{- 27} g {cm}^{2} / s

[or

ℏ = {(1.6 \times 10^{- 33} cm)}^{2}

in geometric units] is very small compared to the "quantities of action" or "magnitudes of the Hamilton-Jacobi function,

S

" or "dynamic phase,

S

" encountered in most everyday applications.
6. It is of no help in localizing the probability distribution that this dynamic phase obeys the simple Hamilton-Jacobi law of propagation,

\begin{matrix} (2) & - \frac{\partial S}{\partial t} = H (\frac{\partial S}{\partial x}, x) = \frac{1}{2 m} {(\frac{\partial S}{\partial x})}^{2} + V (x) . \end{matrix}

It is of no help in localizing the probability distribution that the solution of this equation for a particle of energy $E$ is extraordinarily simple,

\begin{matrix} (3) & S (x, t) = - E t + \int_{x_{0}}^{x} {2 m [E - V (x)]}^{1 / 2} d x + δ_{E} \end{matrix}

(with

δ_{E}

an arbitrary additive phase constant). The probability amplitude is still spread all over everywhere. There is not the slightest trace of anything like a localized world line,

x = x (t)

.
8. To localize the particle, build a probabilityamplitude wave packet by superposing monofrequency (monoenergy) terms, according to a prescription qualitatively of the form

\begin{matrix} (4) & ψ (x, t) = ψ_{E} (x, t) + ψ_{E + Δ E} (x, t) + \dots \cdot \end{matrix}

Superposition of monoenergy waves to give wave packet

Destructive interference takes place almost everywhere. The wave packet is concentrated in the region of constructive interference. There the phases of the various waves agree; thus

\begin{matrix} (5) & S_{E} (x, t) = S_{E + Δ E} (x, t) \end{matrix}

At last one has moved from a wave spread everywhere to a localized wave and thence, in the limit of indefinitely small wavelength, to a classical world line. This one equation of constructive interference ties together

x

and

t

(locus of world line in

x, t

, diagram). Smooth lines

- 20, - 19, - 18

, etc. are wave crests of

ψ_{E}

; dashed lines, wave crests for

ψ_{E + Δ E}

. Shaded area is region of constructive interference (wave packet). Black dots mark locus of classical world line,

{Lim}_{Δ E \to 0} \frac{S_{E + Δ E} (x, t) - S_{E} (x, t)}{Δ E} = 0

9. The Newtonian course of the world line through spacetime follows at once from this condition of constructive interference when one goes to the classical limit (

ℏ

negligible compared to amounts of action involved; hence wavelength negligibly short; hence spread of energies

Δ E

required to build well-localized wave packet also negligible); thus

\frac{S_{E + Δ E} (x, t) - S_{E} (x, t)}{Δ E} = 0

reduces to

\frac{\partial S_{E} (x, t)}{\partial E} = 0

Box 25.3 (continued)

This condition in turn, applied to expression (3), gives the time required to travel to the point

x

; thus,

- t + \int_{x_{0}}^{x} \frac{d x}{{(2 / m) [E - V (x)]}^{1 / 2}} + t_{0} = 0

where

t_{0}

is an abbreviation for the quantity

t_{0} = d δ_{E} / d E

("difference in base value of dynamic phase per unit difference of energy").
10. Not one trace of the quantum of action comes into this final Newtonian result, for a simple reason:

ℏ

has been treated as negligible and the wavelength has been treated as negligible. In this limit the location of the wave "packet" reduces to the location of the wave crest. The location of the wave crest is precisely what is governed by

S_{E} (x, t)

; and the condition of "constructive interference"

\partial S_{E} (x, t) / \partial E = 0

gives without approximation the location of the sharply defined Newtonian world line

x = x (t)

.
11. Relevance in the context of motion in a central field of force? Quickest known route to the concept of effective potential (Box 25.4).

Box 25.4 MOTION UNDER GRAVITATIONAL ATTRACTION OF A CENTRAL MASS ANALYZED BY HAMILTON-JACOBI METHOD

A. Newtonian Theory of Gravitation

Hamiltonian

\begin{matrix} (1) & \tilde{H} = \frac{{\tilde{p}}_{r}^{2}}{2} + \frac{{\tilde{p}}_{θ}^{2}}{2 r^{2}} + \frac{{\tilde{p}}_{ϕ}^{2}}{2 r^{2} \sin^{2} θ} - \frac{M}{r} \end{matrix}

(tildes over energy, momentum, etc., refer to test object of unit mass; test particle of mass

μ

follows same motion with energy

E = μ \tilde{E}

, momentum

p = μ \tilde{p}

, etc.).
Equation of Hamilton-Jacobi for propagation of wave crests:

\begin{matrix} (2) & - \frac{\partial \tilde{S}}{\partial t} = \frac{1}{2} {(\frac{\partial \tilde{S}}{\partial r})}^{2} + \frac{1}{2 r^{2}} {(\frac{\partial \tilde{S}}{\partial θ})}^{2} + \frac{1}{2 r^{2} \sin^{2} θ} {(\frac{\partial \tilde{S}}{\partial ϕ})}^{2} - \frac{M}{r} . \end{matrix}

Box 25.4 (continued)

Solve by "method of separation of variables" with convention that

\sqrt{a^{2}} \equiv \pm a

\begin{aligned} \tilde{S} = & - \tilde{E} t + {\tilde{p}}_{ϕ} ϕ + \int^{θ} {({\tilde{L}}^{2} - \frac{{\tilde{p}}_{ϕ}^{2}}{\sin^{2} θ})}^{1 / 2} d θ \\ (3) & + \int^{r} {[2 (\tilde{E} + \frac{M}{r} - \frac{{\tilde{L}}^{2}}{2 r^{2}})]}^{1 / 2} d r + δ_{{\tilde{p}}_{ϕ}, \tilde{,}, \tilde{E}} . \end{aligned}

(Check by substituting into Hamilton-Jacobi equation. Solution as sum of four terms corresponding to the four independent variables goes hand in hand with expression of probability amplitude in quantum mechanics as product of four factors, because

i S / ℏ = i μ \tilde{S} / ℏ

is exponent in approximate expression for the probability amplitude.)
Constructive interference of waves:
(1) with slightly different

\tilde{E}

values (impose "condition of constructive interference"

\partial {\tilde{S}}_{{\tilde{p}}_{φ}} \tilde{L}, \tilde{E} (t, r, θ, ϕ) / \partial \tilde{E} = 0)

tells when the particle arrives at a given

r

(that is, gives relation between

t

and

r

);
(2) with slightly different values of the "parameter of total angular momentum per unit mass,"

\tilde{L}

(impose condition of constructive interference

\partial {\tilde{S}}_{{\tilde{p}}_{ϕ}, \tilde{L}, \tilde{E}} (t, r, θ, ϕ) / \partial \tilde{L} = 0

) tells correlation between

θ

and

r

(a major feature of the shape of the orbit);
(3) with slightly different values of the "parameter of azimuthal angular momentum per unit mass,"

{\tilde{p}}_{ϕ}

(impose condition

\partial \tilde{S} / \partial {\tilde{p}}_{ϕ} = 0

) gives correlation between

θ

and

ϕ

\begin{matrix} (4) & 0 = \frac{\partial \tilde{S}}{\partial {\tilde{p}}_{ϕ}} = ϕ - \int^{θ} \frac{({\tilde{p}}_{ϕ} / \tilde{L}) d θ}{\sin θ {(\sin^{2} θ - {\tilde{p}}_{ϕ}^{2} / {\tilde{L}}^{2})}^{1 / 2}} \end{matrix}

Planar character of the orbit.
Puzzle out the value of this last integral with the help of a table of integrals? It is quicker and clearer to capture the content without calculation: the particle moves in a plane. The vector associated with the angular momentum

\tilde{L}

stands perpendicular to this plane. The projection of this angular momentum along the

z

-axis is

{\tilde{p}}_{ϕ} = \tilde{L} \cos α

(definition of orbital inclination,

α

). Straight line connecting origin with particle cuts unit sphere in a point

P

. As time runs on,

P

traces out a great circle on the unit sphere. The plane of this great circle cuts the equatorial plane in a "line of nodes," at which "hinge-line" the two planes are separated by a dihedral angle,

α

. The orbit of the point

P

is described by

\hat{x} = r \cos ψ, \hat{y} = r \sin ψ, \hat{z} = 0

in a Cartesian system of coordinates in which

\hat{y}

runs along the line of nodes and in which

\hat{x}

lies in the plane of the orbit.

Box 25.4 (continued)

In a coordinate system in which

y

runs along the line of nodes and

x

lies in the plane of the equator, one has:

\begin{aligned} r \cos θ = z \\ r z \cos α + \hat{x} \sin α = r \cos ψ \sin α; \\ r \sin θ \cos ϕ = x \\ r \sin θ \sin ϕ = y = \hat{z} \sin α + \hat{x} \cos α = r \cos ψ \cos α; \\ r \sin ψ . \end{aligned}

Eliminate reference to the Cartesian coordinates and, by taking ratios, also eliminate reference to

r

. Thus find the equation of the great circle route in parametric form,

\tan ϕ = \tan ψ / \cos α

and

\cos θ = \cos ψ \sin α .

Here increasing values of

ψ

spell out successive points on the great circle. Eliminate

ψ

via the relation

\sec^{2} ψ - \tan^{2} ψ = 1

to find

\frac{\sin^{2} α}{\cos^{2} θ} - \tan^{2} ϕ \cos^{2} α = 1

or, more briefly,

\begin{matrix} (5) & \sec ϕ = \tan α \tan θ . \end{matrix}

One verifies that

ϕ

as calculated from (5) provides an integral of (4), thus confirming the physical argument just traced out. Moreover, the arbitrary constant of integration that comes from (4), left out for the sake of simplicity from (5), is easily inserted by replacing

ϕ

there by

ϕ - ϕ_{0}

(rotation of line of nodes to a new azimuth). The kind of physics just done in tracing out the relation between

θ

and

ϕ

is evidently elementary solid geometry and nothing more. The same geometric relationships also show up, with no relativistic corrections whatsoever (how could there be any?!) for motion in Schwarzschild geometry. Therefore it is appropriate to drop this complication from attention here and hereafter. Let the particle move entirely in the direction of increasing

θ

, not at all in the direction of increasing

ϕ

; that is, let it move in an orbit of zero angular momentum

{\tilde{p}}_{ϕ}

(total angular momentum vector

\tilde{L}

inclined at angle

α = π / 2

z

-axis). Consequently the dynamic phase

S

(to be divided by

ℏ

to obtain phase of Schrödinger wave function when one turns from classical to quantum mechanics) becomes

\begin{matrix} (6) & \tilde{S} = - \tilde{E} t + \tilde{L} θ + \int^{r} {[2 (\tilde{E} + \frac{M}{r} - \frac{{\tilde{L}}^{2}}{2 r^{2}})]}^{1 / 2} d r + δ_{\tilde{L}, \tilde{E}} \end{matrix}

Shape of orbit:

\begin{matrix} (7) & 0 = \frac{\partial \tilde{S}}{\partial \tilde{L}} = θ - \int^{r} \frac{\tilde{L} d r / r^{2}}{{[2 (\tilde{E} + M / r - {\tilde{L}}^{2} / 2 r^{2})]}^{1 / 2}} \end{matrix}

whence

\begin{matrix} (8) & r = \frac{{\tilde{L}}^{2} / M}{1 + e \cos θ} . \end{matrix}

Here

e

is an abbreviation for the eccentricity of the orbit,

\begin{matrix} (9) & e = {(1 + 2 \tilde{E L^{2}} / M^{2})}^{1 / 2} \end{matrix}

(greater than 1 for positive

\tilde{E}

, hyperbolic orbit; equal to 1 for zero

\tilde{E}

, parabolic orbit; less than 1 for negative

\tilde{E}

, elliptic orbit). A constant of integration has been omitted from (8) for simplicity. To reinstall it, replace

θ

θ - θ_{0}

(rotation of direction of principal axis in the plane of the orbit). Other features of the orbit:

\begin{matrix} (10) & \begin{array}{lr} (\binom{semimajor axis of}{orbit when elliptic}) & a = \frac{{\tilde{L}}^{2} / M}{1 - e^{2}} = \frac{M}{(- 2 \tilde{E})}; \\ (\binom{semiminor axis of}{orbit when elliptic}) & b = \frac{{\tilde{L}}^{2} / M}{{(1 - e^{2})}^{1 / 2}} = \frac{\tilde{L}}{(- 2 \tilde{E})^{1 / 2}}; \end{array} \end{matrix}

\begin{matrix} (12) & (\begin{array}{l} (\begin{matrix} "impact parameter" \\ for hyperbolic orbit, \\ or "distance of closest \\ approach in \\ absence of deflection" \end{matrix} \end{array}) b = \frac{(angular momentum per unit mass)}{(linear momentum per unit mass)} \end{matrix}

\begin{aligned} Θ & = π - 2 arc \cos (1 / e) \\ (14) & = 2 \arctan [M / (2 \tilde{E})^{1 / 2} \tilde{L}] \\ = 2 \arctan [M / 2 \tilde{E} b]; \end{aligned}

\begin{matrix} (13) & (\binom{actual distance of}{closest approach}) r_{min} = \frac{{\tilde{L}}^{2} / M}{{(1 + 2 \tilde{E L^{2}} / M^{2})}^{1 / 2} + 1} \end{matrix}

(\binom{angle of deflection}{in hyperbolic orbit})

(\binom{differential scattering}{cross section}) \frac{d σ}{d Ω} = \frac{2 π b d b}{2 π \sin Θ d Θ}

\begin{matrix} (15) & = \frac{M^{2}}{{(4 \tilde{E} \sin^{2} Θ / 2)}^{2}} (Rutherford). \end{matrix}

Box 25.4 (continued)

Time as correlated with position:

\begin{matrix} (16) & 0 = \frac{\partial \tilde{S}}{\partial \tilde{E}} = - t + \int^{r} \frac{d r}{{[2 (\tilde{E} + \frac{M}{r} - \frac{{\tilde{L}}^{2}}{2 r^{2}})]}^{1 / 2}} \end{matrix}

Write

\begin{matrix} (17) & r = \frac{M}{(- 2 \tilde{E})} (1 - e \cos u) \end{matrix}

to simplify the integration. Get

\begin{matrix} (18) & t = \frac{M}{(- 2 \tilde{E})^{3 / 2}} (u - e \sin u), \\ (19) & (\binom{mean circular}{frequency}) = \frac{2 π}{(period)} = ω = \frac{(- 2 \tilde{E})^{3 / 2}}{M} = {(\frac{M}{a^{3}})}^{1 / 2} . \end{matrix}

Here

u

is the so-called "mean eccentric anomaly" (Bessel's time parameter). In terms of this quantity, one has also:

\begin{matrix} \sin u = \frac{{(1 - e^{2})}^{1 / 2} \sin θ}{1 + e \cos θ}; \\ \cos u = \frac{\cos θ + e}{1 + e \cos θ}; \\ \cos θ = \frac{\cos u - e}{1 - e \cos u}; \\ \sin θ = \frac{{(1 - e^{2})}^{1 / 2} \sin u}{1 - e \cos u}; \\ (20) & x = r \cos θ = \frac{M}{(- 2 \tilde{E})} (\cos u - e); \\ (21) & y = r \sin θ = \frac{\tilde{L}}{(- 2 \tilde{E})^{1 / 2}} \sin u . \end{matrix}

These expressions lend themselves to Fourier analysis into harmonic functions of the time, with coefficients that are standard Bessel functions:

\begin{matrix} (22) & J_{n} (z) = \frac{1}{2 π} \int_{- π}^{π} e^{i (z \sin u - n u)} d u \\ (23) & x / a = - \frac{3}{2} e + \sum_{\begin{matrix} k = - \infty \\ k \neq 0 \end{matrix}}^{+ \infty} k^{- 1} J_{k - 1} (k e) \cos k ω t; \\ (24) & y / a = {(1 - e^{2})}^{1 / 2} \sum_{\begin{matrix} k = - \infty \\ k \neq 0 \end{matrix}}^{+ \infty} k^{- 1} J_{k - 1} (k e) \sin k ω t \end{matrix}

[for these and further formulas of this type, see, for example, Wintner (1941), Siegel (1956), and Siegel and Moser (1971)]. Via such Fourier analysis one is in a position to calculate the intensity of gravitational radiation emitted at the fundamental circular frequency

ω

and at the overtone frequencies (see Chapter 36 ).

B. Einstein's Geometric Theory of Gravitation

Connection between energy and momentum for a test particle of rest mass

μ

traveling in curved space,

\begin{matrix} (25) & g^{α β} p_{α} p_{β} + μ^{2} = 0 . \end{matrix}

Gravitation shows up in no way other than in curvature of the geometry, in which the particle moves as free of all "real" force. Refer all quantities to basis of a test object of unit rest mass by dealing throughout with

\tilde{p} = p / μ

. Also write

{\tilde{p}}_{α} = \partial \tilde{S} / \partial x^{α}

. Thus Hamilton-Jacobi equation for propagation of wave crests in Schwarzschild geometry (external field of a star;

§ 23.6

) becomes

\begin{matrix} (26) & \begin{array}{r} - \frac{1}{(1 - 2 M / r)} {(\frac{\partial \tilde{S}}{\partial t})}^{2} + (1 - 2 M / r) {(\frac{\partial \tilde{S}}{\partial r})}^{2} + \frac{1}{r^{2}} {(\frac{\partial \tilde{S}}{\partial θ})}^{2} + \frac{1}{r^{2} \sin^{2} θ} {(\frac{\partial \tilde{S}}{\partial ϕ})}^{2} \\ + 1 = 0 \end{array} \end{matrix}

Solve Hamilton-Jacobi equation. As in Newtonian problem, simplify by eliminating all motion in direction of increasing

ϕ

. Thus set

0 = {\tilde{p}}_{ϕ} = \partial \tilde{S} / \partial ϕ

(dynamic phase independent of

ϕ

) and have

\begin{matrix} (27) & \tilde{S} = - \tilde{E} t + \tilde{L} θ + \int^{r} {[{\tilde{E}}^{2} - (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2} \frac{d r}{(1 - 2 M / r)} \end{matrix}

Find shape of orbit by "principle of constructive interference"; thus,

\begin{matrix} (28) & 0 = \frac{\partial \tilde{S}}{\partial \tilde{L}} = θ - \int^{r} \frac{\tilde{L} d r / r^{2}}{{[{\tilde{E}}^{2} - (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2}} \end{matrix}

[See equation (25.41) and associated discussion in text; also Figure 25.6.]
Find time to get to given

r

by considering "interference of wave crests" belonging to slightly different

\tilde{E}

values:

\begin{matrix} (29) & 0 = \frac{\partial \tilde{S}}{\partial \tilde{E}} = - t + \int^{r} \frac{\tilde{E}}{{[{\tilde{E}}^{2} - (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2}} \frac{d r}{(1 - 2 M / r)} \end{matrix}

[See equation (25.32) and associated discussion in text; also Figure 25.5 and exercise 25.15.]

§25.2. SYMMETRIES AND CONSERVATION LAWS

From symmetries to conservation laws by:
(1) Lagrangian or Hamiltonian approach
(2) Killing-vector approach

Killing vector,

ξ

, defined

In analytic mechanics, one knows that symmetries of a Lagrangian or Hamiltonian result in conservation laws. Exercises 25.1 to 25.4 describe how these general principles are used to deduce, from the symmetries of Schwarzschild spacetime, constants of motion for the trajectories (geodesics) of freely falling particles in the gravitational field outside a star. The same constants of motion are obtained in a different language in differential geometry, where a "Killing vector" is the standard tool for the description of symmetry. This section considers the general question of metric symmetries before proceeding to Schwarzschild spacetime.

Let the metric components

g_{μ ν}

relative to some coordinate basis

d x^{α}

be independent of one of the coordinates

x^{K}

, so

\begin{matrix} (25.1) & \partial g_{μ ν} / \partial x^{α} = 0 for α = K \end{matrix}

This relation possesses a geometric interpretation. Any curve

x^{α} = c^{α} (λ)

can be translated in the

x^{K}

direction by the coordinate shift

Δ x^{K} = ε

to form a "congruent" (equivalent) curve:

x^{α} = c^{α} (λ) for α \neq K and x^{K} = c^{K} (λ) + ε .

Let the original curve run from

λ = λ_{1}

λ = λ_{2}

and have length

L = \int_{λ_{1}}^{λ_{2}} {[g_{μ ν} (x (λ)) (d x^{μ} / d λ) (d x^{ν} / d λ)]}^{1 / 2} d λ .

Then the displaced curve has length

L (ε) = \int_{λ_{1}}^{λ_{2}} {[{g_{μ ν} (x (λ)) + ε \frac{\partial g_{μ ν}}{\partial x^{K}}} (d x^{μ} / d λ) (d x^{ν} / d λ)]}^{1 / 2} d λ .

But the coefficient of

ε

in the integrand is zero. Therefore the length of the new curve is identical to the length of the original curve:

d L / d ε = 0

.
The vector

\begin{matrix} (25.2) & ξ \equiv d / d ε = (\partial / \partial x^{K}) \end{matrix}

provides an infinitesimal description of these length-preserving "translations." It is called a "Killing vector." It satisfies Killing's equation*

\begin{matrix} (25.3) & ξ_{μ; p} + ξ_{p; μ} = 0 \end{matrix}

(condition on the vector field

ξ

necessary and sufficient to ensure that all lengths are preserved by the displacement

ε ξ

). This condition is expressed in covariant form.

Therefore it is enough to establish it in the preferred coordinate system of (25.1) in order to have it hold in every coordinate system. In that preferred coordinate system, the vector field, according to (25.2), has components

ξ^{μ} = δ^{μ}_{K}

Therefore the covariant derivative of this vector field has components

\begin{aligned} ξ_{μ; ν} = g_{μ α} ξ_{; ν}^{α} & = g_{μ α} (\frac{\partial ξ^{α}}{\partial x^{ν}} + Γ_{ν σ}^{α} ξ^{σ}) \\ (25.4) & = g_{μ α} Γ_{ν K}^{α} = Γ_{μ ν K} = \frac{1}{2} (\frac{\partial g_{μ K}}{\partial x^{ν}} + \frac{\partial g_{μ ν}}{\partial x^{K}} - \frac{\partial g_{ν K}}{\partial x^{μ}}) \\ = \frac{1}{2} (g_{μ K, ν} - g_{ν K, μ}) . \end{aligned}

One sees that

ξ_{μ; ν}

is antisymmetric in the labels

μ

and

ν

, as stated in Killing's equation (25.3).

The geometric significance of a Killing vector is spelled out in Box 25.5 .
From Killing's equation,

ξ_{(μ; p)} = 0

, and from the geodesic equation

\nabla_{p} p = 0

for the tangent vector

p = d / d λ

to any geodesic, follows an important theorem: In any geometry endowed with a symmetry described by a Killing vector field

ξ

, motion along any geodesic whatsoever leaves constant the scalar product of the tangent vector with the Killing vector:

\begin{matrix} (25.5) & p_{K} = p \cdot ξ = constant . \end{matrix}

In verification of this result, evaluate the rate of change of the constant

p_{K}

along the course of the typical geodesic (affine parameter

λ

; result therefore as applicable to light rays-with zero lapse of proper time-as to particles); thus,

\begin{matrix} (25.6) & d p_{κ} / d λ = {(p^{μ} ξ_{μ})}_{; ν} p^{ν} = (p_{; ν}^{μ} p^{ν}) ξ_{μ} + p^{(μ} p^{ν} ξ_{[μ; ν]} = 0 \end{matrix}

Turn back from a general coordinate system to the coordinates of (25.1), where the Killing vector field of the symmetry lets itself be written

ξ = \partial / \partial x^{K}

. Then the scalar product of (25.5) becomes constant

\equiv p_{α} ξ^{α} = p_{α} δ^{α}_{K} = p_{K}

. The symmetry of the geometry guarantees the conservation of the

K

-th covariant coordinate-based component of the momentum.

On a timelike geodesic in spacetime, the momentum of a test particle of mass

μ

\begin{matrix} (25.7) & p \equiv d / d λ = μ u = μ d / d τ \end{matrix}

Thus the affine parameter

λ

most usefully employed in the above analysis, when it is concerned with a particle, is not proper time

τ

but rather the ratio

λ = τ / μ

When the metric

g_{μ v}

is independent of a coordinate

x^{K}

, that coordinate is called cyclic, and the corresponding conserved quantity,

p_{K}

, is called the "momentum conjugate to that cyclic coordinate" in a terminology borrowed from nonrelativistic mechanics.

Conservation of

p \cdot ξ

for geodesic motion

Terminology:
"cyclic coordinate," "conjugate momentum"

Box 25.5 KILLING VECTORS AND ISOMETRIES (IIlustrated by a Donut)

A. On a given manifold (e.g., spacetime, or the donut pictured here), in a given coordinate system, the metric components are independent of a particular coordinate

x^{K}

. Example of donut:

\begin{matrix} d s^{2} = b^{2} d θ^{2} + (a - b \cos θ)^{2} d ϕ^{2} \\ g_{μ ν} independent of x^{K} = ϕ . \end{matrix}

B. Translate an arbitrary curve

C

through the infinitesimal displacement

ε ξ \equiv ε (\partial / \partial x^{K}) = ε (\partial / \partial ϕ), ε ≪ 1

to form a new curve

C^{'}

. In coordinate language

C

θ = θ (λ), ϕ = ϕ (λ)

; while

C^{'}

θ = θ (λ), ϕ =

ϕ (λ) + ε

. (Translation of all points through

Δ ϕ = ε

.) Because

\partial g_{μ ν} / \partial ϕ = 0

, the curves

C

and

C^{'}

have the same length (see text).
C. Pick a set of neighboring points

a, B, C, D

; and translate each of them through

ε ξ

to obtain points

a^{'}, B^{'}, C^{'}, Q^{'}

. Since the length of every curve is preserved by this translation, the distances between neighboring points are also preserved:

(

distance between

a^{'}

and

B^{'}) =

(distance between

C

and

B

).
But geometry is equivalent to a table of all infinitesimal distances (see Box 13.1). Thus the geometry of the manifold is left completely unchanged by a translation of all points through

ε ξ

. [This is the coordinatefree version of the statement

\partial g_{μ ν} / \partial ϕ = 0

.] One says that

ξ = \partial / \partial ϕ

is the generator of an "isometry" (or "group of motions") on the manifold.
D. In general (see text), a vector field

ξ (P)

generates an isometry if and only if it satisfies Killing's equation

ξ_{(α; β)} = 0

.
E. If

ξ (P)

generates an isometry (i.e. if

ξ

is a "Killing vector"), then the curves

to which

ξ

is tangent

[ξ = {(\partial P / \partial x^{K})}_{α_{1}, \dots, α_{n}}]

are called "trajectories of the isometry."
F. The geometry is invariant under a translation of all points of the manifold through the same

Δ x^{K}

along these trajectories

[P (x^{K}, α_{1}, \dots, α_{n}) ⟶ P (x^{K} +

Δ x^{K}, α_{1}, \dots, α_{n}

); "finite motion" built up from many "infinitesimal motions"

ε ξ

.]
G. This isometry is described in physical terms as follows. Station a family of observers throughout the manifold. Have each observer report to a central computer (1) all aspects of the manifold's geometry near him, and (2) the distances and directions to all neighboring observers (directions relative to "preferred" directions that are determined by anisotropies in the local geometry). Through each observer's position passes a unique trajectory of the isometry. Move each observer through the same fixed

Δ x^{K}

(e.g.,

Δ x^{K} = 17

) along his trajectory, leaving the manifold itself unchanged. Then have each observer report to the central computer the same geometric information as before his motion. The information received by the computer after the motion will be identical to that received before the motion. There is no way whatsoever, by geometric measurements, to discover that the motion has occurred! This is the significance of an isometry.

Three different trajectories on a donut

Parameter on trajectories is

x^{K} = ϕ

EXERCISES

Exercise 25.1. CONSTANT OF MOTION OBTAINED FROM HAMILTON'S PRINCIPLE

Prove the above theorem of conservation of

p_{K} \equiv p \cdot ξ

from Hamilton's principle (Box 13.3)

\begin{matrix} (25.8) & δ \int \frac{1}{2} g_{μ ν} (x) (d x^{μ} / d λ) (d x^{ν} / d λ) d λ = 0 \end{matrix}

as applied to geodesic paths. Recall: In this action principle,

g_{μ}

, is to be regarded as a known function of position,

x

, along the path; and the path itself,

x^{μ} (λ)

, is to be varied.

Exercise 25.2. SUPER-HAMILTONIAN FORMALISM FOR GEODESIC MOTION

Show that a set of differential equations in Hamiltonian form results from varying

p_{μ}

and

x^{μ}

independently in the variational principle

δ I = 0

, where

\begin{matrix} (25.9) & I = \int (p_{μ} d x^{μ} - H d λ) \end{matrix}

and

\begin{matrix} (25.10) & H \equiv \frac{1}{2} g^{μ ν} (x) p_{μ} p_{v} \end{matrix}

Show that the "super-Hamiltonian"

H

is a constant of motion, and that the solutions of these equations are geodesics. What do the choices

K = + \frac{1}{2}, K = 0, K = - \frac{1}{2} μ^{2}

, or

K = - \frac{1}{2}

mean for the geodesic and its parametrization?

Exercise 25.3. KILLING VECTORS IN FLAT SPACETIME

Find ten Killing vectors in flat Minkowski spacetime that are linearly independent. (Restrict attention to linear relationships with constant coefficients).

Exercise 25.4. POISSON BRACKET AS KEY TO CONSTANTS OF MOTION

ξ

is a Killing vector, show that

p_{K} \equiv ξ^{μ} p_{μ}

commutes (has vanishing Poisson bracket) with the Hamiltonian

K

of the previous problem,

[K, p_{K}] = 0

, so

d p_{κ} / d λ = 0

. (Hint: Use a convenient coordinate system.)

Exercise 25.5. COMMUTATOR OF KILLING VECTORS IS A KILLING VECTOR

Consider two Killing vectors,

ξ

and

η

, which happen not to commute [as differential operators; i.e., the commutator of equations (8.13) does not vanish; consider rotations about perpendicular directions as a case in point]; thus,

[ξ, η] \equiv ζ \neq 0

(a) Show that no single coordinate system can be simultaneously adapted, in the sense of equation (25.1), to both the

ξ

and

η

symmetries (see exercise 9.9).
(b) Let

p_{ξ} = p_{μ} ξ^{μ}, p_{η} = p_{μ} η^{μ}

, and

p_{ξ} = p_{μ} ζ^{μ}

, and derive the Poisson-bracket relationship

[p_{ξ}, p_{η}] = - p_{ξ}

. In a geometry, the symmetries of which are related in this way, show that

p_{ξ}

is also a constant of motion.
(c) In a coordinate system where

ζ = (\partial / \partial x^{K})

, define

K

as in (25.10) and show from

[K, p_{ξ}] = 0

that

ζ

is a Killing vector.

Thus the commutator of two Killing vectors is itself a Killing vector.

Exercise 25.6. EIGENVALUE PROBLEM FOR KILLING VECTORS

Show that any Killing vector satisfies

ξ^{μ}_{; μ} = 0

, and is an eigenvector with eigenvalue

κ = 0

of the equation

\begin{matrix} (25.11) & ξ^{μ; ν}_{; p} + R^{μ}_{ν} ξ^{ν} = κ ξ^{μ} . \end{matrix}

Find a variational principle (Raleigh-Ritz type) for this eigenvalue equation.

§25.3. CONSERVED QUANTITIES FOR MOTION IN SCHWARZSCHILD GEOMETRY

Consider a test particle moving in the Schwarzschild geometry, described by the line element

\begin{matrix} (25.12) & d s^{2} = - (1 - 2 M / r) d t^{2} + \frac{d r^{2}}{(1 - 2 M / r)} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

This expression for the geometry applies outside any spherically symmetric center of attraction of total mass-energy

M

. It makes no difference, for the motion of the particle outside, what the geometry is inside, because the particle never gets there; before it can, it collides with the surface of the star-if the center of attraction is a star, that is to say, a fluid mass in hydrostatic equilibrium. At each point throughout such an equilibrium configuration, the Schwarzschild coordinate

r

exceeds the local value of the quantity

2 m (r)

; see

§ 23.8

. Therefore the Schwarzschild coordinate

R

of the surface exceeds

2 M

. Consequently, expression (25.12) applies outside any equilibrium configuration, no matter how compact

(r > R > 2 M

implies that one need not face the issue of the "singularity" at

r = 2 M

). The more compact the configuration, however, the more of the Schwarzschild geometry the test particle can explore. The ideal limit is not a star in hydrostatic equilibrium. It is a star that has undergone complete gravitational collapse to a black hole. Then (25.12) applies arbitrarily close to

r = 2 M

. This idealization is assumed here ("black hole"), because the analysis can then cover the maximum range of possible situations.

Wherever the test particle lies, and however fast it moves, project that point and project that 3 -velocity radially onto a sphere of some fixed

r

value, say, the unit sphere

r = 1

. The point and the vector together define a point and a vector on the surface of the unit sphere; and they in turn mark the beginning and define the totality of a great circle. As the particle continues on its way, the radial projection of its position will continue to lie on that great circle. To depart from the great circle on one side or the other would be to give preference to the one hemisphere or the other of the unit sphere, contrary to the symmetry of the situation.

Orient the coordinate system so that the radial projection of the orbit coincides with the equator,

θ = π / 2

, of the polar coordinates (see Box 25.4 for the spherical trigonometry of a more general orientation of the orbit, and for eventual specializa-

Why attention focuses on particle orbits around a black hole

Choice of coordinates to make particle orbit lie in "equator,"

θ = π / 2

tion to a polar orbit, in contrast to the equatorial orbit considered here). In polar coordinates as so oriented, the particle has at the start, and continues to have, zero momentum in the

θ

direction; thus,

p^{θ} = d θ / d λ = 0 .

Conserved quantities for particle motion:
(1)

E

(2)

L

(3)

μ

(4)

\tilde{E} \equiv E / μ

(5)

\tilde{L} \equiv L / μ

Effective potential

\tilde{V}

, and equations for orbit when

μ \neq 0

The expression (25.12) for the line element shows that the geometry is unaffected by the translations

t ⟶ t + Δ t, ϕ ⟶ ϕ + Δ ϕ

. Thus the coordinates

t

and

ϕ

are "cyclic." The conjugate momenta

p_{0} \equiv - E

and

p_{ϕ} \equiv \pm L (L \geq 0)

are therefore conserved. This circumstance allows one immediately to deduce the major features of the motion, as follows.
The magnitude of the 4 -vector of energy-momentum is given by the rest mass of the particle,

\begin{matrix} (25.13) & g_{α β} p^{α} p^{β} + μ^{2} = g^{α β} p_{α} p_{β} + μ^{2} = 0 \end{matrix}

\begin{matrix} (25.14) & - \frac{E^{2}}{(1 - 2 M / r)} + \frac{1}{(1 - 2 M / r)} {(\frac{d r}{d λ})}^{2} + \frac{L^{2}}{r^{2}} + μ^{2} = 0 . \end{matrix}

Moreover, one knows from the equivalence principle that test particles follow the same world line regardless of mass. Therefore what is relevant for the motion of particles is not the energy and angular momentum themselves, but the ratios

\begin{matrix} \tilde{E} = E / μ = (\binom{energy per unit}{rest mass}), \\ (25.15) & \tilde{L} = L / μ = (\binom{angular momentum}{per unit rest mass}) . \end{matrix}

Recall also

λ = τ / μ = (\binom{proper time per}{unit rest mass}) .

Then (25.14) becomes an equation for the change of

r

-coordinate with proper time in which the rest mass makes no appearance:

\begin{aligned} {(\frac{d r}{d τ})}^{2} & = {\tilde{E}}^{2} - (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2}) \\ (25.16a) & = {\tilde{E}}^{2} - {\tilde{V}}^{2} (r) \end{aligned}

Here

\begin{matrix} (25.16b) & \tilde{V} (r) = {[(1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2} \end{matrix}

is the "effective potential" mentioned in

§ 25.1

and Figure 25.2 and to be discussed
in

$ 25.5

. For the rate of change of the other two relevant coordinates with proper time, one has, assuming a "direct" orbit

(d ϕ / d τ > 0; p_{ϕ} = + L

rather than

- L)

\begin{matrix} (25.17) & \frac{d ϕ}{d τ} = \frac{1}{μ} \frac{d ϕ}{d λ} = \frac{p^{ϕ}}{μ} = \frac{g^{ϕ ϕ} L}{μ} = \frac{\tilde{L}}{r^{2}} \end{matrix}

and

\begin{matrix} (25.18) & \frac{d t}{d τ} = \frac{1}{μ} \frac{d t}{d λ} = \frac{p^{0}}{μ} = - \frac{g^{00} E}{μ} = \frac{\tilde{E}}{1 - 2 M / r} \end{matrix}

Knowing

r

as a function of

τ

from (25.16), one can find

ϕ

and

t

in their dependence on

τ

from (25.17) and (25.18). Symmetry considerations have in effect reduced the four coupled second-order differential equations

p^{μ}_{; p} p^{ν} = 0

of geodesic motion to the single first-order equation (25.16).

For objects of zero rest mass, it makes no sense to refer to proper time, and a slightly different treatment is appropriate (

$ 25.6

Before looking, in

§ 25.5

, at the motions predicted by equations (25.16) to (25.18), it is useful to analyze the physical significance of the constants

p_{0}

and

p_{ϕ}

, and to identify other physically significant quantities whose values will be of interest in studying these orbits. One calls

E = - p_{0}

the "energy at infinity"; and

L = | p_{ϕ} |

, for equatorial orbits, the "total angular momentum." To justify these names, compare them with standard quantities measured by an observer at rest on the equator of the Schwarzschild coordinate system as the test particle flies past him in its orbit. Let

\begin{aligned} E_{local} & {\equiv p^{\hat{0}} \equiv ⟨ ω^{\hat{0}}, p ⟩ \equiv ⟨ | g_{00} |}^{1 / 2} d t, p ⟩ = {| g_{00} |}^{1 / 2} p^{0} \\ (25.19) & = {| g_{00} |}^{1 / 2} d t / d λ = (1 - 2 M / r)^{1 / 2} d t / d λ \end{aligned}

be the energy he measures in his proper reference frame, and let

\begin{aligned} (25.20) & v_{\hat{ϕ}} & \equiv \frac{p^{\hat{ϕ}}}{p^{\hat{0}}} \equiv \frac{⟨ ω^{\hat{}}, p ⟩}{E_{local}} = \frac{{⟨ | g_{ϕ ϕ} |}^{1 / 2} d ϕ, d / d λ ⟩}{E_{local}} \\ = \frac{r (d ϕ / d λ)}{E_{local}} = \frac{p_{ϕ}}{r E_{local}} \end{aligned}

be the tangential component of the ordinary velocity he measures. [Note:

ω^{\hat{α}}

are the basis one-forms of the observer's proper reference frame; see equations (

23.15 a, b

).] In terms of these locally measured quantities, the energy-at-infinity is

\begin{matrix} (25.21) & E = - p_{0} = - g_{00} p^{0} = {| g_{00} |}^{1 / 2} E_{local} = (1 - 2 M / r)^{1 / 2} E_{local} = constant. \end{matrix}

It therefore represents the locally measured energy

E_{local}

, corrected by a factor

{| g_{00} |}^{1 / 2}

. For any particle that flies freely (geodesic motion) from this observer to

r = \infty

, the correction factor reduces to unity, and

E_{local}

(as measured by a second observer, this time at infinity) becomes identical with

E

. Similarly the angular momentum from (25.20) is

\begin{matrix} (25.22) & p_{ϕ} = E_{local} {\hat{O}}_{\overset{ϕ}{2}} r = constant. \end{matrix}

Interpretation of $E$ as

"energy at infinity" and

L

as "angular momentum"

This, like

E = - p_{0}

, represents a quantity that is conserved, and whose interpretation for

r ⟶ \infty

on any orbit is familiar. Finally, recall that the total 4 -momentum of two colliding particles

p_{1} + p_{2}

{(p_{μ})}_{1} + {(p_{μ})}_{2}

is conserved in a point collision (at any

r

). Therefore the totals

(E)_{1} + (E)_{2} = {(- p_{0})}_{1} + {(- p_{0})}_{2}

and

{(p_{ϕ})}_{1} + {(p_{ϕ})}_{2}

are also conserved. One of the colliding particles may be on an orbit that could never reach out to

r = \infty

, but this makes no difference. This conservation principle allows and forces one to take over the terms

E =

"energy at infinity" and

L =

"angular momentum," valid for orbits that do reach to infinity, for an orbit that does not reach to infinity.

EXERCISES

Exercise 25.7. RADIAL VELOCITY OF A TEST PARTICLE

Obtain a formula for the radial component of velocity

v_{\hat{r}}

that an observer at

r

would measure [see (25.20) for

v_{\hat{ϕ}}

]. Express

E_{local}, v_{\hat{r}}

, and

v_{\hat{ϕ}}

in terms of

r

and the constants

E, p_{ϕ}

Exercise 25.8. ROTATIONAL KILLING VECTORS FOR SCHWARZSCHILD GEOMETRY

(a) Show that in the isotropic coordinates of exercise 23.1, the metric for the Schwarzschild geometry takes the form

\begin{matrix} (25.23) & d s^{2} = - (1 - M / 2 \bar{r})^{2} (1 + M / 2 \bar{r})^{- 2} d t^{2} + (1 + M / 2 \bar{r})^{4} (d {\bar{r}}^{2} + {\bar{r}}^{2} d Ω^{2}) \end{matrix}

(b) Exhibit a coordinate transformation that brings this into the form

\begin{matrix} (25.24) & d s^{2} = - (1 - M / 2 \bar{r})^{2} (1 + M / 2 \bar{r})^{- 2} d t^{2} + (1 + M / 2 \bar{r})^{4} (d x^{2} + d y^{2} + d z^{2}) \end{matrix}

with

\bar{r} = {(x^{2} + y^{2} + z^{2})}^{1 / 2}

.
(c) Show that

ξ_{x} = y (\partial / \partial z) - z (\partial / \partial y)

and similar vectors

ξ_{y}

and

ξ_{z}

are each Killing vectors, by verifying (see exercise 25.5 c ) that the Poisson brackets

[K, L_{K}]

vanish for each

L_{K} = p \cdot ξ_{K}, K = x, y, z

.
(d) Show that

ξ_{z} = (\partial / \partial ϕ)_{t, r, θ}

; and show that for orbits in the equatorial plane

L_{z} = p_{ϕ}

L_{x} = L_{y} = 0

Exercise 25.9. CONSERVATION OF TOTAL ANGULAR MOMENTUM OF A TEST PARTICLE

Prove by a Poisson-bracket calculation that the total angular momentum squared,

L^{2} =

p_{θ}^{2} + (\sin θ)^{- 2} p_{ϕ}^{2}

is a constant of motion for any Schwarzschild geodesic.

Exercise 25.10. SELECTING EQUATION BY SELECTING WHAT IS VARIED

Write out the integral

I

that is varied in (25.8) for the special case of the Schwarzschild metric (25.12). What equation results from the demand

δ I = 0

if only

ϕ (λ)

is varied? If only

t (λ)

Exercise 25.11. MOTION DERIVED FROM SUPER-HAMILTONIAN

Write out the super-Hamiltonian (25.10) for the special case of the Schwarzschild metric. Deduce from its form that

p_{0}

and

p_{ϕ}

are constants of motion. Derive (25.14), (25.17), and (25.18) from this super-Hamiltonian formalism.

§25.4. GRAVITATIONAL REDSHIFT

The conservation law

{| g_{00} |}^{1 / 2} E_{local} =

constant (equation 25.21), which is valid in this form for any time-independent metric with

g_{0 j} = 0

and for particles with both zero and non-zero rest mass, is sometimes called the "law of energy red-shift." It describes how the locally measured energy of any particle or photon changes (is "red-shifted" or "blue-shifted") as it climbs out of or falls into a static gravitational field. For a particle of zero rest mass (photon or neutrino), the locally measured energy

E_{local}

, and wavelength

λ_{local}

(not to be confused with affine parameter!), are related by

E_{local} = h / λ_{local}

, where

h

is Planck's constant. Consequently, the law of energy red-shift can be rewritten as

\begin{matrix} (25.25) & λ_{local} {| g_{o o} |}^{- 1 / 2} = constant. \end{matrix}

A photon emitted by an atom at rest in the gravitational field at radius

r

, and received by an astronomer at rest at infinity is red-shifted by the amount

\begin{aligned} z \equiv Δ λ / λ = (λ_{received} - λ_{emitted}) / λ_{emitted} = {| g_{00} (r) |}^{- 1 / 2} - 1 \\ (25.26) & z = (1 - 2 M / r)^{- 1 / 2} - 1 \\ (25.26~N) & z \approx M / r in Newtonian limit. \end{aligned}

Note that these expressions are valid whether the photon travels along a radial path or not.

Exercise 25.12. REDSHIFT BY TIMED PULSES

EXERCISE

Derive expression (25.26) for the photon redshift by considering two pulses of light emitted successively by an atom at rest at radius

r

. [Hint: If

Δ τ_{em}

is the proper time between pulses as measured by the emitting atom, and

Δ τ_{rec}

is the proper time separation as measured by the observer at

r = \infty

, then one can idealize

λ_{em}

Δ τ_{em}

and

λ_{rec}

Δ τ_{rec.}

Law of "energy redshift" ('gravitational redshift")

Qualitative features of orbits diagnosed from effective-potential diagram

and illustrated in Figure 25.2 and Box 25.6. The first diagram in Box 25.6 gives

{\tilde{V}}^{2} (r)

as a function of

r

. It is relevant even in the "domain inside the black hole"

(r < 2 M)

, where

{\tilde{V}}^{2}

is negative (see Chapter 31). It serves as a model for, and is closely related to, the "effective potential"

B^{- 2} (r)

used in

§ 25.6

to analyze photon orbits. The final diagram in Box 25.6 gives

\tilde{V} (r)

itself as a function of

r

. Energy levels in this diagram or in Figure 25.2 can be interpreted as in any conventional energy-level diagram. The difference in energy between two levels represents energy, as measured at infinity, of the photon given off in the transition from the one level to the other. Whether one plots

\tilde{V} (r)

{\tilde{V}}^{2} (r)

as a function of

r

is largely a matter of convenience. The important point is this: a value of

r

where

\tilde{V} (r)

becomes equal to the available energy

\tilde{E}

, or

{\tilde{V}}^{2} (r)

becomes equal to

{\tilde{E}}^{2}

, is a turning point. A particle that was moving to larger

r

values, once arrived at a turning point, turns around and moves to smaller

r

values. Or when a particle moving to smaller

r

values comes to a turning point, it reverses its motion and proceeds to larger

r

values. A turning point is not a point of equilibrium. A stone thrown straight up does not sit at a point of equilibrium at the top of its flight. However, when

\tilde{E} - \tilde{V} (r)

, or

{\tilde{E}}^{2} - {\tilde{V}}^{2} (r)

, instead of having a single root, has a double root, then one does deal with a point of equilibrium (only possible because of "centrifugal force" fighting against gravity). When this equilibrium occurs at a minimum of

\tilde{V} (r)

, it is a stable equilibrium; at a maximum, an unstable equilibrium. Thus all the major features of the motion in the

r

direction can be read from a plot of the effective potential as a function of

r

(plot depends on value of

\tilde{L}

) and from a knowledge of the

\tilde{E}

value (Figure 25.2, with further details in Box 25.6).

Box 25.6 QUALITATIVE FEATURES OF ORBITS OF A PARTICLE MOVING IN SCHWARZSCHILD GEOMETRY

A. Equations Governing Orbit (see text for derivation)

Effective-potential equation for radial part of motion:

\begin{matrix} (d r / d τ)^{2} + {\tilde{V}}^{2} (\tilde{L}, r) = {\tilde{E}}^{2} \\ {\tilde{V}}^{2} (\tilde{L}, r) = (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2}) \end{matrix}

\tilde{E} =

(energy at infinity per unit rest mass),

\tilde{L} =

(angular momentum per unit rest mass).
2. Supplementary equations for angular and time motion for "direct" orbit,

d \dot{ϕ} / d τ > 0

\begin{matrix} d ϕ / d τ = \tilde{L} / r^{2} \\ \frac{d t}{d τ} = \frac{\tilde{E}}{1 - 2 M / r} \end{matrix}

"Turning points" of orbit occur where horizontal line of height

{\tilde{E}}^{2}

crosses

{\tilde{V}}^{2}

B. Newtonian Limit,

| \tilde{E} - 1 | ≪ 1

M / r ≪ 1, \tilde{L} / r ≪ 1

Speak not about "energy-at-infinity per unit rest mass," $\tilde{E} = E / μ = {(1 - v_{\infty}^{2})}^{- 1 / 2}$ , but instead about the "nonrelativistic energy per unit rest mass,"

ε \equiv \frac{1}{2} ({\tilde{E}}^{2} - 1) \approx \tilde{E} - 1 \approx \frac{1}{2} v_{\infty}^{2} .

Speak not about ${\tilde{V}}^{2} (\tilde{L}, r)$ but instead about the Newtonian effective potential,

V_{N} (\tilde{L}, r) \equiv \frac{1}{2} ({\tilde{V}}^{2} - 1) \approx - \frac{M}{r} + \frac{{\tilde{L}}^{2}}{2 r^{2}} .

Rewrite effective-potential equation in the form

\frac{1}{2} {(\frac{d r}{d τ})}^{2} + V_{N} (\tilde{L}, r) = ε

From the effective-potential diagram and the subsidiary equation $d ϕ / d τ = \tilde{L} / r^{2}$ , conclude that:
a. Particles with $ε \geq 0 (\tilde{E} \geq 1)$ come in from $r = \infty$ along hyperbolic or parabolic orbits, are reflected off the effective potential at $ε = V_{N} [{\tilde{E}}^{2} = {\tilde{V}}^{2}$ ; "turning point"; $(d r / d τ)^{2} = 0]$ , and return to $r = \infty$ .
b. Particles with $ε < 0 (\tilde{E} < 1)$ move back and forth in an effective potential well between periastron (inner turning point of elliptic orbit) and apastron (outer turning point).

C. Relativistic Orbits

Use the effective-potential diagram of part A (reproduced here for various

\overset{L}{L}

), in the same way one uses the Newtonian diagram of part

B

, to deduce the qualitative features of the orbits. The main conclusions are these.

Box 25.6 (continued)

Orbits with periastrons at $r ≫ M$ are Keplerian in form, except for the periastron shift (exercise 25.16; §40.5) familiar for Mercury.
Orbits with periastrons at $r ⩽ 10 M$ differ markedly from Keplerian orbits.
For $\bar{L} / M \leq 2 \sqrt{3}$ there is no periastron; any incoming particle is necessarily pulled into $r = 2 M$ .
For $2 \sqrt{3} < \tilde{L} / M < 4$ there are bound orbits in which the particle moves in and out between periastron and apastron; but any particle coming in from $r = \infty$ (unbound; ${\tilde{E}}^{2} \geq 1$ ) necessarily gets pulled into $r = 2 M$ .
For $L^{†} = \tilde{L} / M > 4$ , there are bound orbits; particles coming in from $r = \infty$ with

\begin{matrix} {\tilde{E}}^{2} < {\tilde{V}}_{max}^{2} = (1 - 2 u_{m}) (1 + L^{† 2} u_{m}^{2}) \\ u_{m} \equiv \frac{1 + \sqrt{1 - 12 / L^{† 2}}}{6} \end{matrix}

reach periastrons and then return to

r =

\infty

; but particles from

r = \infty

with

{\bar{E}}^{2} >

{\tilde{V}}_{max}^{2}

get pulled into

r = 2 M

.
6. There are stable circular orbits at the minimum of the effective potential; the minimum moves inward from

r = \infty

for

\tilde{L} =

\infty

r = 6 M

for

L^{†} = \tilde{L} / M = 2 \sqrt{3}

. The most tightly bound, stable circular orbit

(\tilde{L} / M = 2 \sqrt{3}, r = 6 M)

has a fractional binding energy of

\frac{μ - E}{μ} = 1 - \bar{E} = 1 - \sqrt{8 / 9} = 0.0572

There are unstable circular orbits at the maximum of the effective potential; the maximum moves outward from $r = 3 M$ for $\tilde{L} = \infty$ to $r = 6 M$ for $\tilde{L} / M = 2 \sqrt{3}$ . A particle in such a circular orbit, if perturbed inward, will spiral into $r = 2 M$ . If perturbed outward, and if it has ${\tilde{E}}^{2} > 1$ , it will escape to $r = \infty$ . If perturbed out-

ward, and if it has ${\bar{E}}^{2} < 1$ , it will either reach an apastron and then enter a spiraling orbit that eventually falls into the star (e.g., if $δ \tilde{E} > 0$ , with unchanged angular momentum); or it will move out and in between apastron and periastron, in a stable bound orbit (e.g., if $δ \tilde{E} < 0$ , again with unchanged angular momentum).

When one turns from qualitative features to quantitative results, one finds it appropriate to write down explicitly the proper time

Δ τ

required for the particle to augment its Schwarzschild coordinate by the amount

Δ r

; thus (with the convention that square roots may be negative or positive,

\sqrt{a^{2}} \equiv \pm a

)

\begin{matrix} (25.27) & τ = \int d τ = \int \frac{d r}{{[{\tilde{E}}^{2} - (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2}} \end{matrix}

The integration is especially simple for a particle falling straight in, or climbing straight out, for then the angular momentum vanishes and the integral can be written in an elementary form that applies (with the change

τ ⟶ t

) even in Newtonian mechanics,

\begin{matrix} (25.27') & τ = \int d τ = \int \frac{d r}{[2 M / r - 2 M / R]^{1 / 2}} \end{matrix}

Here

R \equiv 2 M / (1 - E^{2})

is the radius at which the particle has zero velocity ("apastron"). The motion follows the same "cycloid principle" that is so useful in nonrelativistic mechanics (Figure 25.3). Thus, in parametric form, one has

\begin{matrix} r = \frac{R}{2} (1 + \cos η), \\ (25.28) & τ = \frac{R}{2} {(\frac{R}{2 M})}^{1 / 2} (η + \sin η), \end{matrix}

with the total proper time to fall from rest at

r = R

into

r = 0

given by the expression

\begin{matrix} (25.29) & τ = \frac{π}{2} R {(\frac{R}{2 M})}^{1 / 2} \end{matrix}

(shorter by a factor

1 / \sqrt{2}

than the time for fall under pull of the same mass, distributed over a sphere of radius

R

; see dotted curve in Figure 25.3).

What about the Schwarzschild-coordinate time taken for a given motion? Take equation (25.16a) for general motion (radial or nonradial), and where

d r / d τ

appears, replace it by

\begin{matrix} (25.30) & \frac{d r}{d τ} = \frac{d r}{d t} \frac{d t}{d τ} = \frac{d r}{d t} \frac{\tilde{E}}{1 - 2 M / r} = \tilde{E} \frac{d r^{*}}{d t} . \end{matrix}

Here

r^{*}

is an abbreviation for a new "tortoise coordinate,"

\begin{matrix} (25.31) & r^{*} = \int d r^{*} = \int \frac{d r}{1 - 2 M / r} = r + 2 M \ln (\frac{r}{2 M} - 1) \end{matrix}

which was introduced by Wheeler (1955) and popularized by Regge and Wheeler (1957). Thus find the equation

\begin{matrix} (25.32) & {(\tilde{E} \frac{d r^{*}}{d t})}^{2} + {\tilde{V}}^{2} = {\tilde{E}}^{2} \end{matrix}

Figure 25.3.
A cycloid gives the relation between proper time and Schwarschild

r

coordinate for a test particle falling straight in toward center of gravitational attraction of negligible dimensions. The angle of turn of the wheel as it rolls on the base line and generates the cycloid is denoted by

η

. In terms of this parameter, one has

\begin{matrix} r = \frac{R}{2} (1 + \cos η) (Schwarzschild r -coordinate) \\ τ = \frac{R}{2} {(\frac{R}{2 M})}^{1 / 2} (η + \sin η) (proper time) \end{matrix}

(note difference in scale factors in expressions for

r

and for

τ

). The total lapse of proper time to fall from

r = R

r = 0

τ = (π / 2) {(R^{3} / 2 M)}^{1 / 2}

. The same cycloid relation and the same expression for time to fall holds in Newton's nonrelativistic theory of gravitation, except that there the symbol

τ

is to be replaced by the symbol

t

(ordinary time). Were one dealing in Newtonian theory with the same attracting mass

M

spread uniformly over a sphere of radius

R

, with a pipe thrust through it to make a channel for the motion of the test particle, then that particle would execute simple harmonic oscillations (dotted curve above). The angular frequency

ω

of these vibrations would be identical with the angular frequency of revolution of the test particle in a circle just grazing the surface of the planet, a frequency given by Kepler's law

M = ω^{2} R^{3}

. In this case, the time to fall to the center would be

(π / 2) {(R^{3} / M)}^{1 / 2}

, longer by a factor

2^{1 / 2}

than for a concentrated center of attraction (concentrated mass: stronger acceleration and higher velocity in the later phases of the fall). The expression for the Schwarzschild-coordinate time

t

required to reach any point

r

in the fall under the influence of a concentrated center of attraction is complicated and is not shown here (see equation 25.37 and Figure 25.5).

The same cycloidal relation that connects

r

with time for free fall of a particle also connects the radius of the "Friedmann dust-filled universe" with time (see Box 27.1), except that there the cycloid diagram applies directly, without any difference in scale between the two key variables:

(\binom{radius of}{3 -sphere}) = \frac{a}{2} (1 - \cos η) ≃ \frac{a}{4} η^{2} (for small η)

(\begin{array}{l} coordinate time \\ identical with \\ proper time as \\ measured on dust \\ particle \end{array}) = \frac{a}{2} (η - \sin η) ≃ \frac{a}{12} η^{3} (for small η)

The starting point of

η

is renormalized to time of start of expansion; see Lindquist and Wheeler (1957) for more on correlation between fall of particle and expansion of universe.

Here the effective potential is the same effective potential that one dealt with before,

\begin{matrix} (25.33) & \tilde{V} = {[(1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2} \end{matrix}

Moreover, the

\tilde{E}

on the righthand side is the same

\tilde{E}

that appeared in the earlier equation for

(d r / d τ)^{2}

. Therefore the turning points and the qualitative description of the motion are both the same as before. "A turning point is a turning point is
a turning point." Right? Right about turning points; wrong about the conclusion.
The story has it that Achilles never could pass the tortoise. Whenever he caught up with where it had been, it had moved ahead to a new location; and when he got there, it was still further ahead; and so on ad infinitum. Imagine the race between Achilles and the tortoise as running to the left and the expected point of passing as lying at

r = 2 M

. The

r

-coordinate has no inhibition about passing through the value

r = 2 M

. Not so

r^{*}

, the "tortoise coordinate." It can go arbitrarily far in the direction of minus infinity (corresponding to the infinitely many times when Achilles catches up with where the tortoise was) and still

r

remains outside

r = 2 M

$r / 2 M$	1.000001	1.0001	1.01	1.278465	2	5	10	10,000
$r^{*} / 2 M$	-12.8155	-8.2102	-3.5952	0	2	6.386	12.303	$10, 009.210$

It follows that there is a great difference between the description of the motion in terms of the proper time

τ

of a clock on the falling particle (

r

goes all the way from

r = R

down to

r = 0

in the finite proper time of 25.29) and a description of the motion in terms of the Schwarzschild-coordinate time

t

appropriate for the faraway observer (

r^{*}

goes all the way from

r^{*} = R^{*}

down to

r^{*} = - \infty

; infinite

t

required for this; but even in infinite time, as

r^{*}

goes down to

- \infty, r

is only brought asymptotically down to

r \sim 2 M

). Thus the second description of the motion leaves out, and has no alternative but to leave out, the whole range of

r

values from

r = 2 M

down to zero: perfectly good physics, and physics that the falling particle is going to see and explore, but physics that the faraway observer never will see and never can see. If the tortoise coordinate did not exist, it would have to be invented. It invests each factor ten of closer approach to

r = 2 M

with the same interest as the last factor ten and the next to come. It proportions itself in accord with the amount of Schwarzschild-coordinate time available to the faraway observer to study these more and more microscopic amounts of motion in more and more detail.

Figure 25.4 shows the effective potential

\tilde{V}

of (25.33) and of Figure 25.2 replotted as a function of the tortoise coordinate. The approach of

\tilde{V}

to zero at

r = 2 M

shows up as an exponential approach of

\tilde{V}

to zero as

r^{*}

goes to minus infinity. Thus in moving "towards the black hole"

(r = 2 M, r^{*} = - \infty)

, the particle, as described in coordinate time

t

, soon casts off any effective influence of any potential, and moves essentially freely toward decreasing

r^{*}

, in accordance with the equation

\begin{matrix} (25.34) & {(\tilde{E} \frac{d r^{*}}{d t})}^{2} ≃ {\tilde{E}}^{2} \end{matrix}

that is, "with the speed of light"

(d r^{*} / d t ≃ - 1)

. This dependence of

r^{*}

t

implies at once an asymptotic dependence of

r

itself on Schwarzschild-coordinate time

t

, of the form

\begin{matrix} (25.35) & r = 2 M + (constant \times e^{- t / 2 M}) \end{matrix}

This result is independent of the angular momentum of the particle and independent also of the energy, provided only that the energy-per-unit-mass

\tilde{E}

is enough to
(3) details of the approach to the Schwarzschild radius (

r = 2 M

)

Figure 25.4.
Effective potential for motion in Schwarzschild geometry, expressed as a function of the tortoise coordinate, for selected values of the angular momentum of the test particle. The angular momentum

L

is expressed in units

M μ

, where

M

is the mass of the black hole and

μ

the mass of the test particle. The effective potential (including rest mass) is expressed in units

μ

; thus,

\tilde{V} = V / μ

. The tortoise coordinate

r^{*} = r + 2 M \ln (r / 2 M - 1)

is given in units

M

.
surmount the barrier (Figure 25.4) of the effective potential-per-unit-mass

\tilde{V}

. (More will be said on the approach to

r = 2 M

in Chapter 32, on gravitational collapse.)
To replace the asymptotic formula (25.35) by a complete formula requires one to integrate (25.32); thus,

\begin{aligned} (25.36) & t = \int d t & = \int \frac{\tilde{E} d r^{*}}{{[{\tilde{E}}^{2} - {\tilde{V}}^{2}]}^{1 / 2}} \\ = \int \frac{\tilde{E}}{{[{\tilde{E}}^{2} - (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2}} \frac{d r}{(1 - 2 M / r)} . \end{aligned}

The integration here is not easy, even for pure radial motion (

\tilde{L} = 0

), as is seen in the complication of the resulting expression (Khuri 1957):

\begin{aligned} (25.37) & t = [ & (\frac{R}{2} + 2 M) {(\frac{R}{2 M} - 1)}^{1 / 2}] η + \frac{R}{2} {(\frac{R}{2 M} - 1)}^{1 / 2} \sin η \\ + 2 M \ln | \frac{(R / 2 M - 1)^{1 / 2} + \tan (η / 2)}{(R / 2 M - 1)^{1 / 2} - \tan (η / 2)} | \end{aligned}

Here

η

is the same cycloid parameter that appears in equation (25.28) and Figure 25.3 (see the detailed plot in Figure 25.5 of the correlation between

r

and

t

, illustrat-

Figure 25.5.
Fall toward a Schwarzschild black hole as described (a) by a comoving observer (proper time

τ

) and (b) by a faraway observer (Schwarzschild-coordinate time

t

). In the one description, the point

r = 0

is attained, and quickly [see equation (25.28)]. In the other description,

r = 0

is never reached and even

r = 2 M

is attained only asymptotically [equations (25.35) and (25.37)]. The qualitative features of the motion in both cases are most easily deduced by inspection of the "effective potential-per-unit-mass"

\tilde{V}

in its dependence on

r

(Figure 25.2) when one is interested in proper time; or the same effective potential

\hat{V}

in its dependence on the "tortoise coordinate"

r^{*}

[Figure 25.4 and equation (25.31)] when one is interested in Schwarzschild-coordinate time

t

.
ing the asymptotic approach to

r = 2 M

). The difficulty in the integration for

t

, as compared to the ease of the integration for

τ

(25.28), has a simple origin. Only two

r

-values appear in (25.27a) as special points when

\tilde{L}

is zero: the starting point,

r = R

, where the velocity vanishes, and the point

r = 0

, where

d r / d τ

becomes infinite. In contrast (25.36), rewritten as

\begin{matrix} (25.36') & t = \int d t = \int \frac{[1 - 2 M / R]^{1 / 2}}{[2 M / r - 2 M / R]^{1 / 2}} \frac{d r}{(1 - 2 M / r)}, \end{matrix}

contains three special points:

r = R, r = 0

, and the added point with all the new physics,

r = 2 M

. To admit angular momentum is to increase the number of special points still further, and to make the integral unmanageable except numerically or qualitatively (via the potential diagram of Figure 25.4), or in terms of elliptic functions [Hagihara (1931)].

It is often convenient to abstract away from the precise value

r = R

at the start of the collapse. In this event, one deals with the limit

R ⟶ \infty

. Then it is convenient to displace the zero of proper time to the instant of final catastrophe. In this limit, one has

\begin{matrix} τ / 2 M = - (2 / 3) (r / 2 M)^{3 / 2}, \\ (25.38) & t / 2 M = - (2 / 3) (r / 2 M)^{3 / 2} - 2 (r / 2 M)^{1 / 2} + \ln \frac{(r / 2 M)^{1 / 2} + 1}{(r / 2 M)^{1 / 2} - 1} \end{matrix}

At very large negative time, the particle is far away and approaching only very slowly. Then one can write

\begin{matrix} (25.39a) & r = {(9 M τ^{2} / 2)}^{1 / 3} ≃ {(9 M t^{2} / 2)}^{1 / 3} \end{matrix}

whether one refers to coordinate time or to proper time. However, the final stages of infall are again very different, when expressed in terms of proper time (

τ ⟶ 0

r ⟶ 0

), from what they are as expressed in terms of Schwarzschild-coordinate time,

\begin{matrix} (25.39b) & r / 2 M = 1 + 4 e^{- 8 / 3} e^{- t / 2 M} \end{matrix}

Nonradial orbits:
(1) Fourier analysis
(2) details of angular motion

Turning from pure radial motion to motion endowed with angular momentum, one has a situation where one would like to express the principal quantities of the motion (components of displacement, velocity, and acceleration) in Fourier series (in Schwarzschild-coordinate time), these being so convenient in the Newtonian limit in analyzing radiation and perturbations of one orbit by another and tidal perturbations of the moving particle itself by the tide-producing action of the center of attraction. Any exact evaluation of these coefficients would appear difficult. For the time being, the values of the Fourier amplitudes would seem best developed by successive approximations starting from the Newtonian analysis (see Box 25.4 and references cited there).

In connection with any such Fourier analysis, it is appropriate to recall that the fundamental frequency alone appears, and all higher harmonics have zero amplitude, when the motion takes place in an exactly circular orbit (opposite extreme from the pure radial motion of

\tilde{L} = 0

). Therefore it is of interest to note (exercise 25.19) that the circular frequency

ω

of this motion, as measured by a faraway observer, is correlated with the Schwarzschild

r

-value of the orbit by exactly the Keplerian formula of non-relativistic physics:

\begin{matrix} (25.40) & ω^{2} r^{3} = M (exact; general relativity) . \end{matrix}

Turn now from the correlation between

r

and time to the correlation between

r

and angle of revolution (

ϕ

in the analysis here;

θ

in the Hamilton-Jacobi analysis of Box 25.4 ; this difference in name is irrelevant in what follows). Return to equation (25.16),

{(\frac{d r}{d τ})}^{2} + {\tilde{V}}^{2} (r) = {\tilde{E}}^{2}

and recall also equation (25.17)

\frac{d ϕ}{d τ} = \frac{\tilde{L}}{r^{2}} .

Solve the second equation for

d τ

, and substitute into the first to find

\begin{matrix} (25.41) & {(\frac{\tilde{L} d r}{r^{2} d \dot{ϕ}})}^{2} + {\tilde{V}}^{2} (r) = {\tilde{E}}^{2} \end{matrix}

or equivalently, with

u = M / r

and

L^{†} = \tilde{L} / M = L / M μ

\begin{matrix} (25.42) & {(\frac{d u}{d ϕ})}^{2} = \frac{{\tilde{E}}^{2} - (1 - 2 u) (1 + L^{† 2} u^{2})}{L^{† 2}} . \end{matrix}

Exercise 25.16 presents an alternative differential equation derived from this formula, and uses it to obtain the following expression for the angle swept out by the particle or planet, moving in a nearly circular orbit, between two successive points of closest approach:

\begin{matrix} (25.43) & Δ ϕ = \frac{2 π}{{(1 - 6 M / r_{0})}^{1 / 2}} \end{matrix}

The radial motion turns around from ingoing to outgoing, or from outgoing to ingoing, whenever the quantity

{\tilde{E}}^{2} - {\tilde{V}}^{2} (r)

, or

\tilde{E} - \tilde{V} (r)

, plotted as a function of

r

, undergoes a change of sign, and this as clearly here in the correlation between

r

and

ϕ

as in the earlier correlation between

r

and time. Recall again the curves of Figure 25.2 for

\tilde{V} (r)

as a function of

r

for selected

\tilde{L}

values. From them one can read out, without any calculation at all, the principal features of typical orbits (Box 25.6) obtained by detailed numerical calculation. Characteristic features are
(1) circular orbit when

\tilde{E}

coincides with a minimum of the effective potential

\tilde{V} (r)

,
(2) precession when

\tilde{E}

is a little more than

{\tilde{V}}_{min}

,
(3) temporary "orbiting" (many turns around the center of attraction) when

\tilde{E}

is close to a maximum

{\tilde{V}}_{max}

of the effective potential,
(4) "capture into the black hole" when

\tilde{E}

exceeds

{\tilde{V}}_{max}

A more detailed analysis appears in Box 25.6. [For explicit analytic calculation of orbits in the Schwarzschild geometry, see Hagihara (1931), Darwin (1959 and 1961), and Mielnik and Plebanski (1962).]

For orbits of positive energy, no feature of the inverse-square force is better known than the Rutherford scattering formula. It gives the "effective amount of target area" presented by the center of attraction for throwing particles into a faraway receptor that picks up everything coming off into a unit solid angle at a specified angle of deflection

Θ

\begin{matrix} (25.44) & \frac{d σ}{d Ω} = \frac{M^{2}}{{[4 (\tilde{E} - 1) \sin^{2} Θ / 2]}^{2}} (Rutherford; nonrelativistic) \end{matrix}

(derivation in equations 8 to 15 of Box 25.4). When one turns from the Newtonian analysis to the general-relativity treatment, one finds two striking new features of the scattering associated with the phenomenon of orbiting. (1) The particles that come off at a given angle of deflection

Θ

now include not only those that have really been deflected by

Θ

(the only contribution in Rutherford scattering), but also those that have been deflected by

Θ + 2 π, Θ + 4 π, \dots

etc. (an infinite series of contributions). (2) These supplementary contributions, while finite in amount, and even finite in amount "per unit range of

Θ

," are not finite in amount when expressed "per unit of solid angle

d Ω = 2 π \sin Θ d Θ^{''}

in either the forward direction

(Θ = 0)

or the backward direction

(Θ = π)

. This circumstance produces no spectacular change in the forward scattering, for that is already infinite in the nonrelativistic approximation (infinity in Rutherford value of

d σ / d Ω

Θ = 0

is approached, arising from
(3) nearly circular orbits: periastron shift
(4) qualitative features of angular motion

Scattering of incoming particles:
(1) Rutherford (nonrelativistic) cross section
(2) new features due to relativistic gravity
particles flying past with large impact parameters and experiencing small deflections; see exercise 25.21 ). In contrast, the backward scattering, which was perfectly finite in the Rutherford analysis, acquires also an infinity:

\begin{matrix} (25.45) & {(\frac{d σ}{d Ω})}_{θ \sim π} \sim \frac{constant}{\sin Θ} . \end{matrix}

This concentration of scattering in the backward direction is known as a "glory." The effect is most readily seen by looking at the brilliant illumination that surrounds the shadow of one's plane on clouds far below (

180^{\circ}

scattering of light ray within waterdrop). It is also clearly seen in observations on the scattering of atoms by atoms near

Θ = 180^{\circ}

. No dwarf star, not even any neutron star, is sufficiently compact to be out of the way of a high-speed particle trying to make such a

180^{\circ}

turn. Only a black hole is compact enough to produce this effect.
Further interesting features of motion in Schwarzschild geometry appear in the exercises below.

EXERCISES

Exercise 25.13. QUALITATIVE FORMS OF PARTICLE ORBITS

Verify the statements about particle orbits made in part C of Box 25.6 .

Exercise 25.14. IMPACT PARAMETER

For a scattering orbit (i.e., unbound orbit), show that

\tilde{L} = \tilde{E} v_{\infty} b

, where

b

is the impact parameter and

v_{\infty}

the asymptotic ordinary velocity; also show that

\begin{matrix} (25.46) & b = \tilde{L} / {({\tilde{E}}^{2} - 1)}^{1 / 2} \end{matrix}

Draw a picture illustrating the physical significance of the impact parameter.
Exercise 25.15. TIME TO FALL TO

r = 2 M

Show from equation (25.16) and the first picture in Box 25.6 that orbits (general

\tilde{L}

value!) which approach

r = 2 M

do so in a finite proper time, but (equation 25.32) an infinite coordinate time

t

. For equilibrium stars, which must have radii

R > 2 M

, the coordinate time

t

to fall to the surface is finite, of course.

Exercise 25.16. PERIASTRON SHIFT FOR NEARLY CIRCULAR ORBITS
Rewrite equation (25.42) in the form

\begin{matrix} (25.47) & (d u / d ϕ)^{2} + (1 - 6 u_{0}) {(u - u_{0})}^{2} - 2 {(u - u_{0})}^{3} = ({\tilde{E}}^{2} - {\tilde{E}}_{0}^{2}) / L^{† 2} \end{matrix}

Express the constant

u_{0} \equiv M / r_{0}

in terms of

\tilde{L} / M

, and express

{\tilde{E}}_{0}

in terms of

u_{0}

. Show for a nearly circular orbit of radius

r_{0}

that the angle swept out between two successive periastra (points of closest approach to the star) is

\begin{matrix} (25.48) & Δ ϕ = 2 π {(1 - 6 M / r_{0})}^{- 1 / 2} \end{matrix}

Sketch the shape of the orbit for

r_{0} = 8 M

Exercise 25.17. ANGULAR MOTION DURING INFALL

From equation (25.42), deduce that the total angle

Δ ϕ

swept out on a trajectory falling into

r = 0

is finite. The computation is straightforward; but the interpretation, in view of the behavior of

t (λ)

on the same trajectory (equation 25.32 and exercise 25.15 ), is not. The interpretation will be elucidated in Chapter 31.

Exercise 25.18. MAXIMUM AND MINIMUM OF EFFECTIVE POTENTIAL

Derive the expressions given in the caption of Figure 25.2 for the locations of the maximum and the minimum of the effective potential as a function of angular momentum. Determine also the limiting form of the dependence of barrier height on angular momentum in the limit in which

\tilde{L}

is very large compared to

M

Exercise 25.19. KEPLER LAW VALID FOR CIRCULAR ORBITS

From

d ϕ / d τ

of (25.17) and

d t / d τ

of (25.18), deduce an expression for the circular frequency of revolution as seen by a faraway observer; and from the results of exercise 25.18 (or otherwise) show that it fulfills exactly the Kepler relation

ω^{2} r^{3} = M

for any circular orbit of Schwarzschild

r

-value equal to

r

, whether stable (potential minimum) or unstable (potential maximum).

Exercise 25.20. HAMILTON-JACOBI FUNCTION

Construct the locus in the

r, θ

diagram of points of constant dynamic phase

\tilde{S} (t, r, θ) = 0

for

t = 0

and for values

\tilde{L} = 4 M, \tilde{E} = 1

(or for

\tilde{L} = 2 \sqrt{3} M, \tilde{E} = (8 / 9)^{1 / 2}

, or for some other equally simple set of values for these two parameters). Show that the whole set of surfaces of constant

\tilde{S}

can be obtained by rotating the foregoing locus through one angle, then another and another, and recopying or retracing. Interpret physically the principal features of the resulting pattern of curves.

Exercise 25.21. DEFLECTION BY GRAVITY CONTRASTED WITH DEFLECTION BY ELECTRIC FORCE

A test particle of arbitrary velocity

β

flies past a mass

M

at an impact parameter

b

so great that the deflection is small. Show that the deflection is

\begin{matrix} (25.49) & θ = \frac{2 M}{b β^{2}} (1 + β^{2}) \end{matrix}

Derive the deflection according to Newtonian mechanics for a particle moving with the speed of light. Show that (25.49) in the limit

β ⟶ 1

is twice the Newtonian deflection. Derive also (flat-space analysis) the contrasting formula for the deflection of a fast particle of rest mass

μ

and charge

e

by a nucleus of charge

Z e

\begin{matrix} (25.50) & θ = \frac{2 Z e^{2}}{μ b β^{2}} {(1 - β^{2})}^{1 / 2} \end{matrix}

How feasible is it to rule out a "vector" theory of gravitation [see, for example, Brillouin (1970)], patterned after electromagnetism, by observations on the bending of light by the sun? [Hint: To simplify the mathematical analysis, go back to (25.42). Differentiate once with respect to

ϕ

to convert into a second-order equation. Rearrange to put on the left all those terms that would be there in the absence of gravity, and on the right all those that originate from the

- 2 u

term (gravitation) in the factor

(1 - 2 u)

. Neglect the right-hand side of the equation and solve exactly (straight-line motion). Evaluate the perturbing term
on the right as a function of

ϕ

by inserting in it the unperturbed expression for

u (ϕ)

. Solve again and get the deflection.]

Exercise 25.22. CAPTURE BY A BLACK HOLE

Over and above any scattering of particles by a black hole, there is direct capture into the black hole. Show that the cross section for capture is

π b_{crit}^{2}

, with the critical impact parameter

b_{crit}

given by

L_{crit} / {(E^{2} - μ^{2})}^{1 / 2}

. From the formulas in the caption of Fig. 25.2 or otherwise, show that for high-energy particles this cross section varies with energy as

\begin{matrix} (25.51) & σ_{capt} = 27 π M^{2} (1 + \frac{2}{3 {\tilde{E}}^{2}} + \dots) \end{matrix}

(photon limit for

\tilde{E} ⟶ \infty

) and for low energies as

\begin{matrix} (25.52) & σ_{capt} = 16 π M^{2} / β^{2} \end{matrix}

where

β

is the velocity relative to the velocity of light [Bogorodsky (1962)].

§25.6. ORBIT OF A PHOTON, NEUTRINO, OR GRAVITON IN SCHWARZSCHILD GEOMETRY

The concepts of "energy per unit of rest mass" and "angular momentum per unit of rest mass" make no sense for an object of zero rest mass (photon, neutrino, even the graviton of exercise 35.16). However, there is nothing about the motion of such an entity that cannot be discovered by considering the motion of a particle of finite rest mass

μ

and going to the limit

μ ⟶ 0

. In this limit the quantities

\tilde{E} = E / μ

and

\tilde{L} = L / μ

individually go to infinity; but the ratio

\begin{matrix} (25.53) & (\binom{impact para-}{meter}) = b = \frac{(\binom{angular}{momentum})}{(\binom{linear}{momentum})} = \frac{L}{{(E^{2} - μ^{2})}^{1 / 2}} = \frac{\tilde{L}}{{({\tilde{E}}^{2} - 1)}^{1 / 2}} \end{matrix}

goes to the finite value

\begin{matrix} (25.54) & {Lim}_{μ \to 0} \frac{\tilde{L}}{\tilde{E}} = b \end{matrix}

In this limit, equation

(25.41)

for the shape of the orbit reduces at once to the simple form

\begin{matrix} (25.55) & {(\frac{1}{r^{2}} \frac{d r}{d ϕ})}^{2} + \frac{1 - 2 M / r}{r^{2}} = \frac{1}{b^{2}} \end{matrix}

\begin{matrix} (25.56) & {(\frac{1}{r^{2}} \frac{d r}{d ϕ})}^{2} + B^{- 2} (r) = b^{- 2} \end{matrix}

\begin{matrix} (25.57) & {(\frac{d u}{d \dot{ϕ}})}^{2} + u^{2} (1 - 2 u) = {(\frac{M}{b})}^{2} \equiv \frac{1}{{\tilde{b}}^{2}} . \end{matrix}

Whichever way the differential equation for the orbit is written, one term in it depends on the choice of orbit (the term

1 / b^{2}

) the other on the properties of the Schwarzschild geometry, but not on the choice of orbit. This second term defines a kind of effective potential,

\begin{matrix} (25.58) & (\begin{array}{l} "effective \\ potential for \\ photon" \end{array}) \equiv B^{- 2} (r) \equiv \frac{1 - 2 M / r}{r^{2}} \end{matrix}

No attempt is made here to take the square root, as was done for a particle of finite rest mass. There one took the root in order to have a quantity that reduced to the Newtonian effective potential (plus the rest mass) in the nonrelativistic limit; but for light

(v = 1)

there is no nonrelativistic limit. Therefore the effective potential (25.58) is plotted directly in Box 25.7, and used there to analyze some of the principal features of the orbits of a photon in Schwarzschild geometry.

On occasion it has proved useful to plot as a function of

r

, not the "effective potential" of (25.58), but the "potential impact parameter

B (r)

" calculated from that formula [see, for example, Power and Wheeler (1957), Zel'dovich and Novikov (1971)]. This potential impact parameter has the following interpretation: A ray, in order to reach the point

r

, must have an impact parameter

b

that is equal to or less than

B (r)

\begin{matrix} (25.59) & b \leq B (r) ("condition of accessibility"). \end{matrix}

A ray with zero impact parameter (head-on impact), or any impact parameter less than

b_{crit} = min [B (r)] = 3 \sqrt{3} M

, can get to any and all

r

values.

The beautifully simple "effective potential" defined by (25.58) is used in (25.56) to determine the shape of an orbit; that is, the azimuth

ϕ

that the photon has when it gets to a given

r

-value. In other connections, it can be equally interesting to know when, or at what Schwarzschild coordinate time, the photon gets to a given

r

value. More broadly, the geodesic of a photon, for which proper time has no meaning, admits of analysis from first principles by way of an affine parameter

λ

, as contrasted with the device of first considering a particle and then going to the limit

μ ⟶ 0

.
(4) critical impact parameter
(5) affine parameter

Box 25.7 QUALITATIVE ANALYSIS OF ORBITS OF A PHOTON

IN SCHWARZSCHILD GEOMETRY

A. Equations Governing Orbit

Effective-potential equation for radial part of motion:

\begin{aligned} {(\frac{d r}{d λ})}^{2} + B^{- 2} (r) = b^{- 2} \\ B^{- 2} (r) = r^{- 2} (1 - 2 M / r); \\ b = (impact parameter) \end{aligned}

Supplementary equations to determine angular and time motion:

\begin{matrix} d ϕ / d λ = 1 / r^{2} \\ d t / d λ = b^{- 1} (1 - 2 M / r)^{- 1} . \end{matrix}

B. Qualitative Features of Orbits (deduced from effective-potential diagram)

A zero-mass particle with $b > 3 \sqrt{3} M$ , which falls in from $r = \infty$ , is "reflected off the potential barrier" (periastron; $b = B; d r / d λ = 0$ ) and returns to infinity.
a. For $b ≫ 3 \sqrt{3} M$ , the orbit is a straight line, except for a slight deflection of angle $4 M / b$ (exercise $25.21; $ 40.3$ ).
b. For $0 < b - 3 \sqrt{3} M ≪ M$ , the particle circles the star many times ("unstable circular orbit) at $r \approx 3 M$ before flying back to $r = \infty$ .
A zero-mass particle with $b < 3 \sqrt{3} M$ , which falls in from $r = \infty$ , falls into $r = 2 M$ (no periastron).
A zero-mass particle emitted from near $r = 2 M$ escapes to infinity only if it has $b < 3 \sqrt{3} M$ ; otherwise it reaches an apastron and then gets pulled back into $r = 2 M$ .

C. Escape Versus Capture as a Function of Propagation Direction

An observer at rest in the Schwarzschild gravitational field measures the ordinary velocity of a zero-mass particle relative to his orthonormal frame [equations (23.15)]:

\begin{matrix} v_{\hat{r}} = \frac{{| g_{r r} |}^{1 / 2} d r / d λ}{{| g_{00} |}^{1 / 2} d t / d λ} = \pm {(1 - b^{2} / B^{2})}^{1 / 2}; \\ v_{\hat{ϕ}} = \frac{{| g_{ϕ ϕ} |}^{1 / 2} d ϕ / d λ}{{| g_{00} |}^{1 / 2} d t / d λ} = b / B; \\ {(v_{\hat{r}})}^{2} + {(v_{\hat{ϕ}})}^{2} = 1; \end{matrix}

\begin{aligned} δ & \equiv (angle between propagation direction and radial direction) \\ = \cos^{- 1} v_{\hat{r}} = \sin^{- 1} v_{\hat{ϕ}} . \end{aligned}

To be able to cross over the potential barrier, the particle must have

b < 3 \sqrt{3} M

, or

v_{\hat{ϕ}}^{2} B^{2} < 27 M^{2}

, or

\sin^{2} δ < 27 M^{2} / B^{2}

. This result, restated:

A particle of zero rest mass at $r < 3 M$ will eventually escape to infinity, rather than be captured by a black hole at $r = 2 M$ if and only if $v_{r}$ is positive and

\sin δ < 3 \sqrt{3} M B^{- 1} (r)

A particle of zero rest mass at $r > 3 M$ will eventually escape to infinity if and only if: (1) $v_{r}$ is positive, or (2) $v_{r}$ is negative and

\sin δ > 3 \sqrt{3} M B^{- 1} (r)

Return to the statement of the conservation laws (25.17) and (25.18) in the form that makes reference to the affine parameter

λ

but no reference to the rest mass

μ

; thus

\begin{matrix} (25.60) & \frac{d ϕ}{d λ} = \frac{L}{r^{2}} \end{matrix}

and

\begin{matrix} (25.61) & \frac{d t}{d λ} = \frac{E}{1 - 2 M / r} \end{matrix}

Recall that the course of a photon in a gravitational field is governed by its direction but not by its energy. Therefore neither

E

nor

L

individually are relevant but only their ratio, the impact parameter

b = L / E

of (25.54) and exercise 25.14. This circumstance leads one to replace the affine parameter

λ

by a new affine parameter,

\begin{matrix} (25.62) & λ_{new} = L λ, \end{matrix}

that is equally constant along the world line of the photon. In this notation (drop the subscript "new" hereafter), the conservation laws take the form

\begin{matrix} (25.63) & \frac{d ϕ}{d λ} = \frac{1}{r^{2}} \\ (25.64) & \frac{d t}{d λ} = \frac{1}{b (1 - 2 M / r)} \end{matrix}

The statement that the world line of the photon is a line of zero lapse of proper time,

\begin{matrix} (25.65) & g_{α β} \frac{d x^{α}}{d λ} \frac{d x^{β}}{d λ} = 0 \end{matrix}

leads to the "radial equation"

\begin{matrix} (25.66) & {(\frac{d r}{d λ})}^{2} + B^{- 2} (r) = b^{- 2} \end{matrix}

Here one encounters again the "effective potential"

B^{- 2} (r)

(25.58)

. The present fuller set of equations for the geodesic of a photon have the advantage that they reach beyond space to a description of the world line in spacetime.
Return to space! Figure 25.6 shows typical orbits for a photon in Schwarzschild geometry. Figure 25.7 shows angle of deflection as a function of impact parameter.
(7) scattering cross section From the information contained in this curve, one can evaluate the contributions to the differential scattering cross section

\begin{matrix} (25.67) & \frac{d σ}{d Ω} = \sum_{"branches"} | \frac{2 π b d b}{2 π \sin Θ d Θ} | \end{matrix}

from the various "branches" of the scattering curve of Figure 25.7 [one turn around the center of attraction, two turns, etc.; for more on these branches and the central

Figure 25.6.
The orbit of a photon in the "equatorial plane" of a black hole, plotted in terms of the Schwarzschild coordinates

r

and

ϕ

, for selected values of the turning point of the orbit,

r_{TP} / M = 2.99, 3.00

(unstable circular orbit),

3.01, 3.5, 4, 5, 6, 7, 8, 9

. The impact parameter is given by the formula

b = r_{TP} {(1 - 2 M / r_{TP})}^{- 1 / 2}

. In none of the cases shown, even for the inward plunging spiral, is the impact parameter less than

b_{crit} = (27)^{1 / 2} M

, nor are any of these orbits able to cross the circle

r = 3 M

. That only happens for orbits with

b

less than

b_{crit}

. For such orbits there is no turning point; the photon comes in from infinity and ends up at

r = 0

: straight in for

b = 0

(head-on impact); only after many loops near

r = 3 M

, when

b / M = (27)^{1 / 2} - ε

, where

ε

is a very small quantity. Appreciation is expressed to Prof. R. H. Dicke for permission to publish these curves, which he had a digital calculator compute and plot out directly from the formula

d^{2} u / d ϕ^{2} =

3 u^{2} - u

, where

u = M / r

.
role of the deflection function

Θ = Θ (b)

in the analysis of scattering, see, for example, Ford and Wheeler (1959a,b)]. For small angles the "Rutherford" part of the scattering predominates. The major part of the small-angle scattering, and in the limit

Θ ⟶ 0

all of it, comes from large impact parameters, for which one has

\begin{matrix} (25.68) & Θ = \frac{4 M}{b} \end{matrix}

(see exercises 25.21 and 25.24 ). It follows that the limiting form of the cross section is

\begin{matrix} (25.69) & \frac{d σ}{d Ω} = {(\frac{4 M}{Θ^{2}})}^{2} (small Θ) \end{matrix}

Also, at

Θ = π

one has a singularity in the differential scattering cross section, with the character of a glory [see discussion following equation (25.44)]. Writing down the contributions of the several branches of the scattering function to the differential cross section, and summing them, one has, near

Θ = π

\begin{matrix} (25.70) & \frac{d σ}{d Ω} = \frac{M^{2}}{π - Θ} (1.75 + 0.0029 + 0.0000055 + \dots) = 1.75 \frac{M^{2}}{π - Θ} . \end{matrix}

Thus, in principle, if one shines a powerful source of light onto a black hole, one gets a direct return of a few photons from it. Equation (25.70) provides a means to calculate the strength of this return. See exercise 25.26.

Figure 25.7.
Deflection of a photon by a Schwarzschild black hole, or by any spherically symmetric center of attraction small enough not to block the trajectory of the photon. The accurate calculations (smooth curves) are compared with formulas (dashed curves) valid asymptotically in the two limiting cases of an impact parameter,

b :

(1) very close to

b_{crit} = 3^{3 / 2} M

(many turns around the center of attraction); and (2) very large compared to

b_{erit}

(small deflection). The algorithm for the accurate calculation of the deflection proceeds as follows (all distances being given, for simplicity, in units of the mass value,

M

). (1) Choose a value,

r = R

, for the Schwarzschild coordinate of the point of closest approach. (2) Calculate the impact parameter,

b

, from

b^{2} = R^{3} / (R - 2)

. (3) Calculate

Q

from

Q^{2} = (R - 2) (R + 6)

. (4) Determine the modulus,

k

, of an "elliptic integral of the first kind" from

\sin^{2} θ = k^{2} = (Q - R + 6) / 2 Q

. (5) Determine the so-called amplitude

ϕ = ϕ_{min}

of the same elliptic function from

{sn}^{2} u_{min} = \sin^{2} {\dot{ϕ}}_{min} =

(2 + Q - R) / (6 + Q - R)

. (6) Then the total deflection is

Θ = 4 (R / Q)^{1 / 2} [F (π / 2, θ) - F (ϕ_{min}, θ)] - π

The values plotted here were kindly calculated by James A. Isenberg on the basis of the work of C. G. Darwin (1959, 1961).
(8) gravitational lens effect

When the source of illumination, instead of being on the observer's side of the black hole, is on the opposite side, then in addition to the "lens effect" experienced by photons flying by with large impact parameter [literature too vast to summarize here, but see, e.g., Refsdal (1964)], and subsumed in equation (25.68), there is a glory type of illumination (intensity

\sim 1 / \sin Θ

, with now, however,

Θ

close to zero) received from photons that have experienced deflections

Θ = 2 π, 4 π, \dots

. This illumination comes from "rings of brightness" located at impact parameters given by

b / M - 3^{3 / 2} = 0.0065, 0.000012, \dots

. Interesting though all these optical effects are as matters of principle, they are, among all the ways to observe a black hole, the worst; see part VI, C, of Box 33.3 for a detailed discussion.

Exercise 25.23. QUALITATIVE FEATURES OF PHOTON ORBITS

EXERCISES

Verify all the statements about orbits for particles of zero rest mass made in Box 25.7 .

Exercise 25.24. LIGHT DEFLECTION

Using the dimensionless variable

u = M / r

in place of

r

itself, and

u_{b} = M / b

in place of the impact parameter, transform (25.55) into the first-order equation

\begin{matrix} (25.71) & {(\frac{d u}{d ϕ})}^{2} + (1 - 2 u) u^{2} = u_{b}^{2} \end{matrix}

and thence, by differentiation, into

\begin{matrix} (25.72) & \frac{d^{2} u}{d ϕ^{2}} + u = 3 u^{2} \end{matrix}

(a) In the large-impact-parameter or small-

u

approximation, in which the term on the right is neglected, show that the solution of

(25.72)

yields elementary rectilinear motion (zero deflection).
(b) Insert this zero-order solution into the perturbation term

3 u^{2}

on the righthand side of (25.72), and solve anew for

u

("rectilinear motion plus first-order correction"). In this way, verify the formula for the bending of light by the sun given by putting

β = 1

in equation (25.49).

Exercise 25.25. CAPTURE OF LIGHT BY A BLACK HOLE

Show that a Schwarzschild black hole presents a cross section

σ_{capt} = 27 π M^{2}

for capture of light.

Exercise 25.26. RETURN OF LIGHT FROM A BLACK HOLE

Show that flashing a powerful pulse of light onto a black hole leads in principle to a return from rings of brightness located at

b / M - 3^{3 / 2} = 0.151, 0.00028, \dots

. How can one evaluate the difference in time delays of these distinct returns? Show that the intensity

I

of the return (erg

/ {cm}^{2}

) as a function of the energy

E_{0} (erg /

steradian) of the original pulse, the mass

M (cm)

of the black hole, the distance

R

to it, and the lateral distance

r

from the "flashlight" to the receptor of returned radiation is

I = \frac{E_{0}}{R^{3} r} \sum_{θ = (2 N + 1) π} | \frac{2 b d b}{d Θ} | = \frac{E_{0} M^{2}}{R^{3} r} (1.75 + 0.0029 + 0.0000055 + \dots)

under conditions where diffraction can be neglected.

§25.7. SPHERICAL STAR CLUSTERS

By combining orbit theory, as developed in this chapter, with kinetic theory in curved spacetime as developed in

§ 22.6

, one can formulate the theory of relativistic star clusters.

Consider, for simplicity, a spherically symmetric cluster of stars (e.g., a globular cluster, but one so dense that relativistic gravitational effects might be important).
(1) foundations for analysis
(2) solution of Vlasoff equation

Demand that the cluster be static, in the sense that the number density in phase space

R

is independent of time. (New stars, flying along geodesic orbits, enter a fixed region in phase space at the same rate as "old" stars leave it.) Ignore collisions and close encounters between stars; i.e., treat each star's orbit as a geodesic in the spherically symmetric spacetime of the cluster as a whole.

With these idealizations accepted, one can write down a manageable set of equations for the structure of the cluster.* Since the cluster is static and spherical, so must be its gravitational field. Consequently, one can introduce the same kind of coordinate system ("Schwarzschild coordinates") as was used for a static spherical star in Chapter 23:

\begin{matrix} (25.73) & d s^{2} = - e^{2 ϕ} d t^{2} + e^{2 Λ} d r^{2} + r^{2} d Ω^{2}; Φ = Φ (r), Λ = Λ (r) \end{matrix}

In the tangent space at each event in spacetime reside the momentum vectors of the swarming stars. For coordinates in this tangent space ("momentum space"), it is convenient to use the physical components of 4 -momentum,

p^{\hat{α}}

-i.e., components on the orthonormal frame

\begin{matrix} (25.74) & ω^{\hat{t}} = e^{ϕ} d t, ω^{\hat{r}} = e^{Λ} d r, ω^{\hat{θ}} = r d θ, ω^{\hat{ϕ}} = r \sin θ d ϕ \end{matrix}

Then the number density of stars in phase space is a spherically symmetric, static function

\begin{matrix} (25.75) & R = R [r, p^{\hat{0}}, p^{\hat{r}}, {(p^{\hat{θ} 2} + p^{\hat{ϕ} 2})}^{1 / 2}] . \end{matrix}

[

∵

is independent of

t

because the cluster is static; and independent of

θ, ϕ

, and angle

Θ = \tan^{- 1} (p^{\hat{ϕ}} / p^{\hat{θ}})

because of spherical symmetry.]

The functions describing the structure of the cluster,

Φ, Λ

, and

π

, are determined by the kinetic (also, in this context, called the Vlasoff) equation (

$ 22.6

)

\begin{matrix} (25.76a) & \begin{array}{r} d R / d λ = 0, i.e., R conserved along orbit \\ of each star in phase space; \end{array} \end{matrix}

and by the Einstein field equations

\begin{matrix} (25.76b) & G^{\hat{α} \hat{β}} = 8 π T^{\hat{α} \hat{β}} = 8 π \int (ϰ p^{\hat{α}} p^{\hat{β}}) μ^{- 1} d p^{\hat{o}} d p^{\hat{r}} d p^{\hat{θ}} d p^{\hat{ϕ}} \end{matrix}

[The Vlasoff equation for Newtonian star clusters is treated by Ogorodnikov (1965). The above expression for the stress-energy tensor of a swarm of particles (stars) was derived in exercise 22.18. Here, as in exercise 22.18, the particles (stars) are assumed not all to have the same rest mass. Note that rest mass is here denoted

μ

, but in Chapter 22 it was denoted

m

.]
To solve the Vlasoff equation, one need only note that

R

is conserved along stellar orbits and therefore must be a function of the constants of the orbital motion. There is a constant of motion corresponding to each Killing vector in the cluster's static, spherical spacetime (see exercise 25.8 ):

E = "energy at infinity" = - p \cdot (\partial / \partial t) = - p_{0},

L_{z} = " z -component of angular momentum" = p \cdot ξ_{z} = p \cdot (\partial / \partial ϕ) = p_{ϕ}

\begin{matrix} (25.77a) & L_{y} = " y -component of angular momentum" = p \cdot ξ_{y}, \end{matrix}

$ L_{x} = $ " $ x $ - c o m p o n e n t o f a n g u l a r m o m e n t u m " $ = p \cdot ξ_{x} $ . I n a d d i t i o n, e a c h s t a r^{'} s r e s t m a s s

\begin{matrix} (25.77b) & μ = {(p^{\hat{0} 2} - p^{\hat{r} 2} - p^{\hat{θ} 2} - p^{\hat{ϕ} 2})}^{1 / 2} \end{matrix}

i s a c o n s t a n t o f i t s m o t i o n . T h e g e n e r a l s o l u t i o n o f t h e V l a s o f f e q u a t i o n, t h e n, h a s t h e f o r m

\mathscr{T}=H\left(E, L_{x}, L_{y}, L_{z}, \mu\right)

But this general solution is not spherically symmetric. For example, the distribution function

ℜ = H (E, μ, L_{z}) δ (L_{y}) δ (L_{x}),

corresponds to a cluster of stars with orbits all in the equatorial plane

θ =

π / 2

(

L_{y} = L_{x} = 0

for all stars in cluster). To be spherical the cluster's distribution function must depend only on the magnitude

L = {(L_{x}^{2} + L_{y}^{2} + L_{z}^{2})}^{1 / 2}

of the angular momentum, and not on its direction (not on the orientation of a star's orbital plane). Thus, the general spherical solution to the Vlasoff equation in a static, spherical spacetime must have the form

\begin{matrix} (25.78) & r = F (E, L, μ) . \end{matrix}

To use this general solution, one must reexpress the constants of the motion

E

L, μ

, in terms of the agreed-on phase-space coordinates

(t, r, θ, ϕ, p^{\hat{0}}, p^{\hat{γ}}, p^{\hat{θ}}, p^{\hat{ϕ}})

. The rest mass of a star is given by (25.77b). The energy-at-infinity is obtained by redshifting the locally measured energy

\begin{matrix} (25.79a) & E = - p_{0} = e^{ϕ} p^{\hat{0}} \end{matrix}

For an orbit in the equatorial plane

(p_{θ} = p^{θ} = p^{\hat{θ}} = 0; L_{x} = L_{y} = 0)

, the total angular momentum has the form

L = | L_{z} | = | p_{ϕ} | = | r p^{\hat{ϕ}} | = r \times ("tangential" component of 4-momentum).

By symmetry, the equation

L = r \times

("tangential" component of

p

) must hold true also for orbits in other planes; it must be perfectly general:

\begin{matrix} (25.79b) & L = r p^{\hat{r}}, \end{matrix}

p^{\hat{r}} \equiv (

tangential component of 4-momentum

) = {[{(p^{\hat{θ}})}^{2} + {(p^{\hat{ϕ}})}^{2}]}^{1 / 2}

(see exercise 25.9).
(3) "smeared-out" stress-energy tensor due to stars

Before solving the Einstein field equations, one finds it useful to reduce the stress-energy tensor to a more explicit form than (25.76b). The off-diagonal components

T^{\hat{0} \hat{j}}

and

T^{\hat{j} \hat{k}} (j \neq k)

all vanish because their integrands are odd functions of

p^{\hat{i}}

. The integrands for the diagonal components

T^{\hat{0} \hat{0}}, T^{\hat{r}}

, and

\frac{1}{2} (T^{\hat{θ} \hat{θ}} + T^{\hat{ϕ} \hat{ϕ}})

are independent of angle

Θ \equiv \tan^{- 1} (p^{\hat{ϕ}} / p^{\hat{θ}})

in the tangential momentum plane; so the momentum volume element can be rewritten as

d p^{\hat{0}} d p^{\hat{r}} d p^{\hat{θ}} d p^{\hat{ϕ}} ⟶ 2 π p^{\hat{r}} d p^{\hat{r}} d p^{\hat{r}} d p^{\hat{0}}

Changing variables from (

p^{\hat{r}}, p^{\hat{r}}, p^{\hat{γ}}

) to

(p^{\hat{r}}, μ, p^{\hat{0}})

where

μ = {[{(p^{\hat{θ}})}^{2} - {(p^{\hat{ı}})}^{2} - {(p^{\hat{T}})}^{2}]}^{1 / 2},

and recognizing that two values of

p^{\hat{r}} (\pm p^{\hat{r}})

correspond to each value of

μ

, one brings the volume element into the form

2 π p^{\hat{r}} d p^{\hat{r}} d p^{\hat{r}} d p^{\hat{0}} ⟶ 4 π (p^{\hat{r}} μ / p^{\hat{r}}) d p^{\hat{r}} d p^{\hat{0}} d μ

The diagonal components of

T

[equation (25.76b)] then read

\begin{aligned} (25.81a) & ρ \equiv T^{\hat{0} \hat{0}} = (total density of mass-energy) \\ = 4 π \int F (e^{ϕ} p^{\hat{0}}, r p^{\hat{r}}, μ) (p^{\hat{0} 2} p^{\hat{r}} / p^{\hat{r}}) d p^{\hat{r}} d p^{\hat{0}} d μ, \\ (25.81b) & P_{T} \equiv \frac{1}{2} (T^{\hat{\hat{θ}} \hat{θ}} + T^{\hat{ϕ} \hat{ϕ}}) = T^{\hat{\hat{θ}} \hat{θ}} = T^{\hat{ϕ} \hat{ϕ}} = (tangential pressure) \\ = 2 π \int F (e^{ϕ} p^{\hat{0}}, r p^{\hat{r}}, μ) [{(p^{\hat{r}})}^{3} / p^{\hat{r}}] d p^{\hat{r}} d p^{\hat{o}} d μ, \\ P_{r} \equiv T^{\hat{r} \hat{r}} = (radial pressure) \\ (25.81c) & = 4 π \int F (e^{ϕ} p^{\hat{0}}, r p^{\hat{r}}, μ) (p^{\hat{γ}} p^{\hat{r}}) d p^{\hat{r}} d p^{\hat{0}} d μ . \end{aligned}

When performing these integrals, one must express

p^{\hat{γ}}

in terms of the variables of integration,

\begin{matrix} (25.81d) & p^{\hat{r}} = {[{(p^{\hat{\hat{}}})}^{2} - {(p^{\hat{t}})}^{2} - μ^{2}]}^{1 / 2} \end{matrix}

The Einstein field equations for this stress-energy tensor and the metric (25.73), after use of expressions (14.43) for

G^{\hat{α} \hat{β}}

and after manipulations analogous to those for a spherical star (

$ 23.5

), reduce to

\begin{matrix} (25.82a) & e^{2 Λ} = (1 - 2 m / r)^{- 1}, m = \int_{0}^{r} 4 π r^{2} ρ d r; \\ (25.82b) & \frac{d Φ}{d r} = \frac{m + 4 π r^{3} P_{r}}{r (r - 2 m)} . \end{matrix}

These equations, together with the assumed form

F (E, L, μ)

of the distribution
function and the integrals (25.81) for

ρ, P_{r}

, and

P_{T}

, determine the structure of the cluster. Box 25.8 gives an overview of these structure equations, and specializes them for an isotropic velocity distribution. Box 25.9 presents and discusses the solution to the equations for an isothermal star cluster (truncated Maxwellian velocity distribution).

Exercise 25.27. ISOTROPIC STAR CLUSTER

EXERCISES

For a cluster with distribution function independent of angular momentum, derive properties B. 1 to B. 6 of Box 25.8 .

Exercise 25.28. SELF-SIMILAR CLUSTER [See Bisnovatyi-Kogan and Zel'dovich (1969), Bisnovatyi-Kogan and Thorne (1970).]
(a) Find a solution to the equations of structure for a spherical star of infinite central density, with the equation of state

P = γ ρ

, where

γ

is a constant

(0 < γ < 1 / 3)

.
(b) Find an isotropic distribution function

F (E, μ)

that leads to a star cluster with the same distributions of

ρ, P, m

, and

Φ

as in the gas sphere of part (a). (See Box 25.8.) [Answer:

\begin{matrix} P = γ ρ = \frac{γ^{2}}{1 + 6 γ + γ^{2}} \frac{1}{2 π r^{2}}, \\ e^{2 A} = (1 - 2 m / r)^{- 1} = (1 + 6 γ + γ^{2}) / (1 + γ)^{2}, \\ e^{2 Φ} = B r^{4 γ / (1 + γ)}, B = const; \\ F = A {(E / B^{1 / 2})}^{- (1 + γ) / γ} δ (μ - μ_{0}) = A r^{- 2} {(E_{local})}^{- (1 + γ) / γ}, A = const.] \end{matrix}

Exercise 25.29. CLUSTER WITH CIRCULAR ORBITS

What must be the form of the distribution function to guarantee that all stars move in circular orbits? Specialize the equations of structure to this case. Analyze the stability of the orbits of individual stars in the cluster, using an effective-potential diagram. What conditions must the distribution function satisfy if all orbits are to be stable? [See Einstein (1939), Zapolsky (1968).]

Box 25.8 EQUATIONS OF STRUCTURE FOR A SPHERICAL STAR CLUSTER

A. To Build a Model for a Star Cluster, Proceed as Follows

Specify the distribution function $= F (E, L, μ)$ , where

\begin{aligned} E = energy-at-infinity of a star \\ L = angular momentum of a star, \\ μ = rest mass of a star. \end{aligned}

Solve the following two integro-differential equations for the metric functions $m = \frac{1}{2} r (1 - e^{- 2 Λ})$ and $Φ$ of the line element

Box 25.8 (continued)

\begin{matrix} d s^{2} = - e^{2 Φ} d t^{2} + e^{2 Λ} d r^{2} + r^{2} d Ω^{2} \\ m = \int_{0}^{r} 4 π r^{2} ρ d r \\ \frac{d Φ}{d r} = \frac{m + 4 π r^{3} P_{r}}{r (r - 2 m)} \end{matrix}

where

\begin{matrix} ρ = 4 π \int F (e^{ϕ} p^{\hat{0}}, r p^{\hat{r}}, μ) [{(p^{\hat{o}})}^{2} p^{\hat{r}} / p^{\hat{r}}] d p^{\hat{r}} d p^{\hat{0}} d μ, \\ P_{T} = 2 π \int F (e^{Φ} p^{\hat{0}}, r p^{\hat{r}}, μ) [{(p^{\hat{r}})}^{3} / p^{\hat{r}}] d p^{\hat{r}} d p^{\hat{0}} d μ, \\ P_{r} = 4 π \int F (e^{ϕ} p^{\hat{0}}, r p^{\hat{r}}, μ) (p^{\hat{r}} p^{\hat{r}}) d p^{\hat{r}} d p^{\hat{\hat{}}} d μ, \\ p^{\hat{r}} = {[{(p^{\hat{o}})}^{2} - {(p^{\hat{r}})}^{2} - μ^{2}]}^{1 / 2} . \end{matrix}

The integrations for

ρ, P_{T}

, and

P_{r}

go over all positive

p^{\hat{r}}, p^{\hat{0}}, μ

for which

{(p^{\hat{δ}})}^{2} - {(p^{\hat{T}})}^{2} - μ^{2} \geq 0

B. If the Distribution Function is Independent

of Angular Momentum, Then

$F = F (E, μ)$ .
The distribution of stellar velocities at each point in the cluster is isotropic.
$ρ = 4 π \int F (e^{ϕ} p^{\hat{θ}}, μ) {[{(p^{\hat{θ}})}^{2} - μ^{2}]}^{1 / 2} {(p^{\hat{θ}})}^{2} d p^{\hat{θ}} d μ$ .
The pressure is isotropic:

P_{r} = P_{T} \equiv P \equiv \frac{4 π}{3} \int F (e^{ϕ} p^{\hat{0}}, μ) {(p^{\hat{0} 2} - μ^{2})}^{3 / 2} d p^{\hat{0}} d μ

The total density of mass-energy $ρ$ , the pressure $P$ , and the metric functions $ϕ$ and $m = \frac{1}{2} r (1 - e^{- 2 Λ})$ satisfy the equations of structure for a gas sphere ("star"),

\begin{matrix} m = \int 4 π r^{2} ρ d r \\ \frac{d Φ}{d r} = \frac{m + 4 π r^{3} P}{r (r - 2 m)}, \\ \frac{d P}{d r} = - \frac{(ρ + P) (m + 4 π r^{3} P)}{r (r - 2 m)} . \end{matrix}

Thus, to every static, spherical star cluster with isotropic velocity distribution, there corresponds a unique gas sphere that has the same distributions of $ρ, P, m$ , and $Φ$ .
Conversely [see Fackerell (1968)], given a gas sphere (solution to equations of stellar structure for $ρ, P, m$ , and $Φ$ ), one can always find a distribution function $F (E, μ)$ that describes a cluster with the same $ρ, P, m$ , and $Φ$ . But for some gas spheres $F$ is necessarily negative in part of phase space, and is thus unphysical.

Box 25.9 ISOTHERMAL STAR CLUSTERS

A. Distribution Function

In any relativistic star cluster, one might expect that occasional close encounters between stars would "thermalize" the stellar distribution function. This suggests that one study isotropic, spherical clusters with the Boltzmann distribution function (tacitly assumed zero for $p^{\hat{0}} = E e^{- ϕ} < μ_{0}$ )

\begin{matrix} (1) & ϰ = F (E, L, μ) = K e^{- E / T} δ (μ - μ_{0}) . \end{matrix}

Here

K

is a normalization constant,

T

is a constant "temperature," and for simplicity the stars are all assumed to have the same rest mass

μ_{0}

.
2. In such a cluster, an observer at radius

r

sees a star of energy-at-infinity

E

to have locally measured energy

\begin{matrix} (2) & p^{\hat{0}} = (rest mass-energy) + (kinetic energy) = \frac{μ_{0}}{{(1 - v^{2})}^{1 / 2}} = E e^{- Φ (r)} . \end{matrix}

Consequently, the stars in his neighborhood have a Boltzmann distribution

\begin{matrix} (3) & \frac{d N}{d^{3} \hat{p} d^{3} \hat{x} d μ} = π = K \exp (- p^{\hat{o}} / T_{loc}) δ (μ - μ_{0}) \end{matrix}

with locally measured temperature

\begin{matrix} (4) & T_{loc} (r) = T e^{- ϕ (r)} . \end{matrix}

Thus, the temperature of the cluster is subject to identically the same red-shift-blueshift effects as photons, particles, and stars that move about in the cluster. (For a derivation of this same temperature-redshift law for a gas in thermal equilibrium, see part (e) of exercise 22.7.)
3. Actually, the Boltzmann distribution (1) can never be achieved. Stars with

E > μ_{0}

are gravitationally unbound from the cluster and will escape. The Boltzmann distribution presumes that, as such stars go zooming off toward

r = \infty

, an equal number of stars with the same energies come zooming in from

r = \infty

to maintain an unchanged distribution function. Such a situation is clearly unrealistic. Instead, one expects the escape of stars to truncate the distribution at some energy

E_{max}

slightly less than

μ_{0}

. The result, in idealized form, is the "truncated Boltzmann distribution"

\begin{matrix} (5) & ϰ = F (E, L, μ) = {\begin{array}{cc} K e^{- E / T} δ (μ - μ_{0}), & E < E_{max}, \\ 0, & E > E_{max} \end{array} \end{matrix}

Box 25.9 (continued)

B. Structure and Stability of Cluster Models

Models for star clusters with truncated Boltzmann distributions have been constructed by Zel'dovich and Podurets (1965), by Fackerell (1966), and by Ipser (1969), using the procedure of Box 25.8. Ipser has analyzed the collisionless radial vibrations of such clusters.
In general, these clusters form a 4-parameter family ( $K, T, μ_{0}, E_{max}$ ). Replace the parameter $K$ by the total rest mass of the cluster, $M_{0} = μ_{0} N$ , where $N$ is the total number of stars. Replace $T$ by the temperature per unit rest mass, $\tilde{T} = T / μ_{0}$ . Replace $E_{max}$ by the maximum energy per unit rest mass, ${\tilde{E}}_{max} = E_{max} / μ_{0}$ . Then the clusters are parametrized by $(M_{0}, \tilde{T}, μ_{0}, {\tilde{E}}_{max})$ . When one now doubles $μ_{0}$ , holding $M_{0}, \tilde{T}, {\tilde{E}}_{max}$ fixed (and thus halving the total number of stars), all macroscopic features of the cluster remain unchanged. In this sense $μ_{0}$ is a "trivial parameter" and can henceforth be ignored or changed at will. The total rest mass of the cluster $M_{0}$ can be regarded as a "scaling factor"; all dimensionless features of the cluster are independent of it. For example, if $ρ_{c}$ is the central density of mass-energy [equation (25.81a), evaluated at $r = 0$ ], then $ρ_{c} M_{0}^{2}$ is dimensionless and is thus independent of $M_{0}$ , which means that $ρ_{c} \propto M_{0}^{- 2}$ . Only two nontrivial parameters remain: $\tilde{T}$ and ${\tilde{E}}_{max}$ .
Consider as an instructive special case [Zel'dovich and Podurets (1965)] the one-parameter sequence with ${\tilde{E}}_{max} = 1 - \frac{1}{2} \tilde{T}$ . The following figure, computed by Ipser (1969), plots for this sequence the fractional binding energy,

\begin{matrix} (6) & E_{bind} / M_{0} \equiv (M_{0} - M) / M_{0} \end{matrix}

(here

M

is total mass-energy); the square of the angular frequency for collisionless vibrations (vibration amplitude

\propto e^{- i ω t}

) divided by central density of mass-energy,

ω^{2} / ρ_{c}

; and the redshift,

z_{c}

, of photons emitted from the center of the cluster and received at infinity. All these quantities are dimensionless, and thus depend only on the choice of

\tilde{T} = T / μ_{0}

.
4. Notice that all models beyond the point of maximum binding energy

(z_{c} \geq 0.5)

are unstable against collisionless radial perturbations (

ω

imaginary; amplitude of perturbation

\propto e^{| ω | t})

. When perturbed slightly, such clusters must collapse to form black holes. (See Chapter 26 for an analysis of the analogous instability in stars).
5. These results suggest an idealized story of the evolution of a spherical cluster [Zel'dovich and Podurets (1965); Fackerell, Ipser, and Thorne (1969)]. The

cluster would evolve quasistatically along a sequence of spherical equilibrium configurations such as those of the figure. The evolution would be driven by stellar collisions and by the evaporation of stars. When two stars collide and coalesce, they increase the cluster's rest mass and hence its fractional binding energy. When a star gains enough energy from such encounters to escape from the cluster, it carries away excess kinetic energy, leaving the cluster more tightly bound. Thus, both collisions and evaporation should drive the cluster toward states of tighter and tighter binding. When the cluster reaches the point, along its sequence, of maximum fractional binding energy, it can no longer evolve quasistatically. Relativistic gravitational collapse sets in: the stars spiral inward through the gravitational radius of the cluster toward its center, leaving behind a black hole with, perhaps, some remaining stars orbiting it.

It is tempting to speculate that violent events in the nuclei of some galaxies and in quasars might be associated with the onset of such a collapse, or with encounters between an already collapsed cluster (black hole) and surrounding stars.

*For more detailed treatments of this subject see, e.g., Stueckelberg and Wanders (1953), Kluitenberg and de Groot (1954), Meixner and Reik (1959), and references cited therein; see also the references on hydrodynamics cited at the beginning of $$ 22.3$ , and the references on kinetic theory cited at the beginning of $$ 22.6$ .
*For more detailed treatments of this subject see, e.g., Ehlers (1961), Taub (1971), Ellis (1971), Lichnerowicz (1967), Cattaneo (1971), and references cited therein; see also the references on kinetic theory cited at the beginning of $$ 22.6$ .
*Exercise supplied by John M. Stewart.
- Based in part on notes prepared by William L. Burke at Caltech in 1968. For more detailed treatments of geometric optics in curved spacetime, see, e.g., Sachs (1961), Jordan, Ehlers, and Sachs (1961), and Robinson (1961); also references discussed and listed in $$ 41.11$ .
*The equations for $A$ are linear. Therefore the analysis would proceed equally well assuming, instead of an amplitude independent of $λ$ , a dominant term $a \propto λ^{n}$ , with $b \propto λ^{n + 1}, c \propto λ^{n + 2}$ , etc. The results are independent of $n$ . Choosing $n = 1$ would give field strengths $F_{μ ν}$ and energy densities $T_{μ ν} \propto F^{2} \propto$ $A^{2} / λ^{2} \propto$ constant as $λ ⟶ 0$ .
*For more detailed and sophisticated treatments of this topic, see, e.g., Tauber and Weinberg (1961), and Lindquist (1966), Marle (1969), Ehlers (1971), Stewart (1971), Israel (1972), and references cited therein. Ehlers (1971) is a particularly good introductory review article.
- Of course, equation (23.5) only succeeds in defining a new time coordinate $t^{'}$ if it is integrable as a differential equation for $t^{'}$ . By choosing the integrating factor $e^{ϕ}$ to be just $e^{ϕ} = a (r)$ , one sees that $t^{'} = t + \int [b (r) / a (r)] d r$ is the integral of (23.5); thus the required $t^{'}$ coordinate always exists, no matter what the functions $a (r), b (r), c (r)$ , and $R (r)$ in equation (23.4) may be.
*Historical note: Wilhelm K. J. Killing, born May 10, 1847, in Burbach, Westphalia, died February 11, 1923 in Münster, Westphalia; Professor of Mathematics at the University of Münster, 1892-1920. The key article that gives the name "Killing vector" to the kind of isometries considered here appeared almost a century ago [Killing (1892)].
*These equations were first derived and explored by Zel'dovich and Podurets (1965).

gravitation6-214bbc04-e7e9-47a7-95a9-eb77e413b5ce

снартев 22

THERMODYNAMICS, HYDRODYNAMICS, ELECTRODYNAMICS, GEOMETRIC OPTICS, AND KINETIC THEORY

§22.1. THE WHY OF THIS CHAPTER

§22.2. THERMODYNAMICS IN CURVED SPACETIME*

Box 22.1 PRINCIPAL ALTERNATIVES FOR "PRIMARY THERMODYNAMIC POTENTIAL" TO DESCRIBE A FLUID

§22.3. HYDRODYNAMICS IN CURVED SPACETIME*

Box 22.2 THERMODYNAMICS AND HYDRODYNAMICS FOR A SIMPLE PERFECT FLUID IN CURVED SPACETIME

A. Ten Quantities Characterize the Fluid

Exercise 22.1. DIVERGENCE OF FLOW LINES PRODUCES VOLUME CHANGES

EXERCISES

Exercise 22.2. EQUATION OF CONTINUITY

Exercise 22.3. CHEMICAL POTENTIAL FOR IDEAL FERMI GAS

Exercise 22.4. PROJECTION TENSORS

Exercise 22.5. PRESSURE GRADIENT IN STATIONARY GRAVITATIONAL FIELD

Exercise 22.6. EXPANSION, ROTATION, AND SHEAR

Exercise 22.7. HYDRODYNAMICS WITH VISCOSITY AND HEAT FLOW.*

§22.4. ELECTRODYNAMICS IN CURVED SPACETIME

Exercise 22.8. THE VECTOR POTENTIAL FOR ELECTRODYNAMICS

Exercise 22.9. CHARGE CONSERVATION IN THE PRESENCE OF GRAVITY

Exercise 22.10. INTERACTING ELECTROMAGNETIC FIELD AND CHARGED MATTER

§22.5. GEOMETRIC OPTICS IN CURVED SPACETIME*

Exercise 22.11. ELECTROMAGNETIC FIELD AND STRESS ENERGY

Exercise 22.12. POLARIZATION

EXERCISES

A. Geometric Optics Assumption

B. Rays, Phase, and Wave Vector (see Box 22.3)

C. Amplitude and Polarization Vector

D. Vector Potential, Electromagnetic Field, and Stress-Energy-Momentum

Box 22.4 (continued)

E. Photon Reinterpretation of Geometric Optics

Exercise 22.13. THE AREA OF A bundle of rays

Exercise 22.14. FOCUSING THEOREM

§22.6. KINETIC THEORY IN CURVED SPACETIME*

Box 22.5 VOLUME IN PHASE SPACE

A. For Swarm of Identical Particles with Nonzero Rest Mass

B. For Swarm of Identical Particles with Zero Rest Mass

Box 22.6 CONSERVATION OF VOLUME IN PHASE SPACE

EXERCISES

Exercise 22.15. INVERSE SQUARE LAW FOR FLUX

Exercise 22.16. BRIGHTNESS OF THE SUN

Exercise 22.17. BLACK BODY RADIATION

Exercise 22.18. STRESS-ENERGY TENSOR

Exercise 22.19. KINETIC THEORY FOR NONIDENTICAL PARTICLES

RELATIVISTIC STARS

CHAPTER 3

SPHERICAL STARS

§23.1. PROLOG

§23.2. COORDINATES AND METRIC FOR A STATIC, SPHERICAL SYSTEM

Exercise 23.1. ISOTROPIC COORDINATES AND NEWTONIAN LIMIT

EXERCISE

§23.3. PHYSICAL INTERPRETATION OF SCHWARZSCHILD COORDINATES

§23.4. DESCRIPTION OF THE MATTER INSIDE A STAR

(1)

EXERCISE

Exercise 23.2. PROPER REFERENCE FRAMES OF FLUID ELEMENTS

§23.5. EQUATIONS OF STRUCTURE

Equation for Λ Λ Lambda\LambdaΛ derived

Box 23.1 MASS-ENERGY INSIDE RADIUS r r rrr

Box 23.1 (continued)

EXERCISES

Exercise 23.4. EINSTEIN CURVATURE TENSOR

Exercise 23.5. TOTAL NUMBER OF BARYONS IN A STAR

Exercise 23.6. BUOYANT FORCE IN A STAR

Exercise 23.7. GRAVITATIONAL ENERGY OF A NEWTONIAN STAR

§23.6. EXTERNAL GRAVITATIONAL FIELD

§23.7. HOW TO CONSTRUCT A STELLAR MODEL

Line Element

Mass Equation

OV Equation of Hydrostatic Equilibrium

Equations of State

Source Equation for Φ Φ Phi\PhiΦ

Exercise 23.8. NEWTONIAN STARS OF UNIFORM DENSITY

EXERCISE

Box 23.2 RELATIVISTIC MODEL STAR OF UNIFORM DENSITY

Box 23.2 (continued)

Box 23.2 (continued)

§23.8. THE SPACETIME GEOMETRY FOR A STATIC STAR

Exercise 23.9. GOOD BEHAVIOR OF r

EXERCISES

Equation for $Λ$ derived

Box 23.1 MASS-ENERGY INSIDE RADIUS $r$

Source Equation for $Φ$

Box 25.1 MASS FROM MEAN ANGULAR FREQUENCY AND SEMIMAJOR AXIS: $M = ω^{2} a^{3}$

Interpretation of $E$ as